Toniolo and Linder, Equation (10+)

Percentage Accurate: 52.9% → 87.0%

Time: 16.5s

Alternatives: 16

Speedup: 2.0×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-189}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-189)
    (/ 2.0 (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.12e-189) {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / l);
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.12e-189) {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.12e-189)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / l));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.12e-189], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.12000000000000002e-189

   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 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. Step-by-step derivation

      1. associate-*l/ 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in t around 0 67.4%

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

   if 1.12000000000000002e-189 < t

   1. Initial program 61.4%

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

   3. Simplified 61.5%

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

      1. associate-/r* 55.7%

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

      2. metadata-eval 55.7%

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

      3. associate-+r+ 55.7%

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

      4. associate-*l* 55.7%

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

      5. associate-*l* 61.4%

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

      6. add-cube-cbrt 61.3%

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

      7. pow3 61.3%

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

   6. Applied egg-rr 79.8%

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

      1. pow1/3 68.0%

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

      2. associate-*l* 68.6%

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

      3. unpow-prod-down 44.3%

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

      4. pow1/3 53.2%

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

   8. Applied egg-rr 53.2%

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

      1. unpow1/3 91.8%

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

   10. Simplified 91.8%

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

Alternative 2: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \end{array} \]

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 5.5e-116)
    (/ 2.0 (/ (* (pow k 2.0) (* (/ t_m l) (/ (pow (sin k) 2.0) (cos k)))) l))
    (/
     2.0
     (*
      (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))))))

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 <= 5.5e-116) {
		tmp = 2.0 / ((pow(k, 2.0) * ((t_m / l) * (pow(sin(k), 2.0) / cos(k)))) / l);
	} else {
		tmp = 2.0 / ((sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	}
	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 <= 5.5e-116) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * ((t_m / l) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	}
	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 <= 5.5e-116)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(t_m / l) * Float64((sin(k) ^ 2.0) / cos(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0)) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	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, 5.5e-116], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.4999999999999998e-116

   1. Initial program 51.1%

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

   3. Simplified 55.3%

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

      1. associate-*l/ 56.3%

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

   6. Applied egg-rr 56.3%

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

   7. Taylor expanded in t around 0 66.9%

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

      1. associate-/l* 66.6%

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

      2. times-frac 68.6%

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

   9. Simplified 68.6%

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

   if 5.4999999999999998e-116 < t

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

      1. associate-/r* 70.9%

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

      2. add-cube-cbrt 70.8%

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

      3. associate-*l* 70.8%

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

      4. associate-/r* 66.9%

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

      5. cbrt-div 66.8%

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

      6. rem-cbrt-cube 66.8%

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

      7. cbrt-prod 70.8%

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

      8. pow2 70.8%

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

      9. pow2 70.8%

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

      10. associate-/r* 66.8%

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

      11. cbrt-div 66.7%

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

      12. rem-cbrt-cube 75.8%

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

      13. cbrt-prod 89.8%

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

      14. pow2 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. unpow2 89.8%

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

      2. cube-mult 89.8%

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

   8. Simplified 89.8%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.8e-117)
    (/ 2.0 (/ (* (pow k 2.0) (* (/ t_m l) (/ (pow (sin k) 2.0) (cos k)))) l))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
      (* (sin k) (pow (/ (pow t_m 1.5) 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 <= 6.8e-117) {
		tmp = 2.0 / ((pow(k, 2.0) * ((t_m / l) * (pow(sin(k), 2.0) / cos(k)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6.8d-117) then
        tmp = 2.0d0 / (((k ** 2.0d0) * ((t_m / l) * ((sin(k) ** 2.0d0) / cos(k)))) / l)
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0))) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.8e-117) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * ((t_m / l) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6.8e-117:
		tmp = 2.0 / ((math.pow(k, 2.0) * ((t_m / l) * (math.pow(math.sin(k), 2.0) / math.cos(k)))) / l)
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (math.pow((k / t_m), 2.0) + 1.0))) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / 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 <= 6.8e-117)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(t_m / l) * Float64((sin(k) ^ 2.0) / cos(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6.8e-117)
		tmp = 2.0 / (((k ^ 2.0) * ((t_m / l) * ((sin(k) ^ 2.0) / cos(k)))) / l);
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (((k / t_m) ^ 2.0) + 1.0))) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-117], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.80000000000000069e-117

   1. Initial program 51.1%

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

   3. Simplified 55.3%

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

      1. associate-*l/ 56.3%

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

   6. Applied egg-rr 56.3%

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

   7. Taylor expanded in t around 0 66.9%

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

      1. associate-/l* 66.6%

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

      2. times-frac 68.6%

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

   9. Simplified 68.6%

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

   if 6.80000000000000069e-117 < t

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

      1. associate-/r* 70.9%

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

      2. add-sqr-sqrt 70.9%

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

      3. pow2 70.9%

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

      4. associate-/r* 66.9%

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

      5. sqrt-div 66.9%

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

      6. sqrt-pow1 75.9%

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

      7. metadata-eval 75.9%

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

      8. sqrt-prod 48.5%

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

      9. add-sqr-sqrt 89.1%

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

   6. Applied egg-rr 89.1%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{t\_m}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 2e-116)
    (/ 2.0 (/ (* (pow k 2.0) (* (/ t_m l) (/ (pow (sin k) 2.0) (cos k)))) l))
    (if (<= t_m 5.2e+137)
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
        (* (sin k) (* (/ t_m l) (/ (pow t_m 2.0) l)))))
      (/ 2.0 (/ (* 2.0 (/ (pow (* t_m (pow (cbrt k) 2.0)) 3.0) l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-116) {
		tmp = 2.0 / ((pow(k, 2.0) * ((t_m / l) * (pow(sin(k), 2.0) / cos(k)))) / l);
	} else if (t_m <= 5.2e+137) {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * (sin(k) * ((t_m / l) * (pow(t_m, 2.0) / l))));
	} else {
		tmp = 2.0 / ((2.0 * (pow((t_m * pow(cbrt(k), 2.0)), 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-116) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * ((t_m / l) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / l);
	} else if (t_m <= 5.2e+137) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * (Math.sin(k) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))));
	} else {
		tmp = 2.0 / ((2.0 * (Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-116)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(t_m / l) * Float64((sin(k) ^ 2.0) / cos(k)))) / l));
	elseif (t_m <= 5.2e+137)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * Float64(sin(k) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0) / l)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-116], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e+137], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2e-116

   1. Initial program 51.1%

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

   3. Simplified 55.3%

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

      1. associate-*l/ 56.3%

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

   6. Applied egg-rr 56.3%

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

   7. Taylor expanded in t around 0 66.9%

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

      1. associate-/l* 66.6%

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

      2. times-frac 68.6%

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

   9. Simplified 68.6%

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

   if 2e-116 < t < 5.1999999999999998e137

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. unpow3 64.0%

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

      2. times-frac 88.9%

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

      3. pow2 88.9%

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

   6. Applied egg-rr 88.9%

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

   if 5.1999999999999998e137 < t

   1. Initial program 72.2%

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

   3. Simplified 58.0%

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

      1. associate-*l/ 58.0%

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

   6. Applied egg-rr 58.0%

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

   7. Taylor expanded in k around 0 58.0%

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

      1. add-cube-cbrt 58.0%

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

      2. pow3 58.0%

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

      3. *-commutative 58.0%

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

      4. cbrt-prod 58.0%

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

      5. rem-cbrt-cube 62.2%

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

      6. unpow2 62.2%

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

      7. cbrt-prod 82.9%

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

      8. pow2 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 8000000000000:\\ \;\;\;\;\frac{2}{\left(k \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \end{array} \]

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 1.6e-92)
    (/ 2.0 (/ (* 2.0 (/ (pow (* t_m (pow (cbrt k) 2.0)) 3.0) l)) l))
    (if (<= k 8000000000000.0)
      (/ 2.0 (* (* k (pow (* t_m (pow (cbrt l) -2.0)) 3.0)) (* 2.0 k)))
      (/
       2.0
       (/ (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* l (cos k))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.6e-92) {
		tmp = 2.0 / ((2.0 * (pow((t_m * pow(cbrt(k), 2.0)), 3.0) / l)) / l);
	} else if (k <= 8000000000000.0) {
		tmp = 2.0 / ((k * pow((t_m * pow(cbrt(l), -2.0)), 3.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (l * cos(k))) / 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 tmp;
	if (k <= 1.6e-92) {
		tmp = 2.0 / ((2.0 * (Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0) / l)) / l);
	} else if (k <= 8000000000000.0) {
		tmp = 2.0 / ((k * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / (((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.6e-92)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0) / l)) / l));
	elseif (k <= 8000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))) / 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_] := N[(t$95$s * If[LessEqual[k, 1.6e-92], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8000000000000.0], N[(2.0 / N[(N[(k * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.5999999999999998e-92

   1. Initial program 56.6%

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

   3. Simplified 58.2%

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

      1. associate-*l/ 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in k around 0 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow3 54.4%

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

      3. *-commutative 54.4%

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

      4. cbrt-prod 54.4%

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

      5. rem-cbrt-cube 63.6%

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

      6. unpow2 63.6%

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

      7. cbrt-prod 71.5%

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

      8. pow2 71.5%

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

   9. Applied egg-rr 71.5%

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

   if 1.5999999999999998e-92 < k < 8e12

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in k around 0 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.5%

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

      3. add-sqr-sqrt 34.1%

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

      4. associate-*l* 34.2%

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

      5. sqrt-div 26.3%

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

      6. sqrt-pow1 26.3%

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

      7. metadata-eval 26.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      8. sqrt-div 26.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      9. sqrt-pow1 26.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      10. metadata-eval 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. associate-*r/ 26.2%

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

       2. *-rgt-identity 26.2%

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

       3. associate-/l/ 26.2%

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

   11. Simplified 26.2%

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

   12. Taylor expanded in k around 0 68.7%

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

       1. associate-/l* 68.7%

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

   14. Simplified 68.7%

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

       1. rem-cube-cbrt 68.5%

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

       2. cbrt-div 68.5%

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

       3. rem-cbrt-cube 69.3%

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

       4. unpow2 69.3%

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

       5. cbrt-prod 83.0%

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

       6. unpow2 83.0%

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

       7. div-inv 83.0%

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

       8. unpow-prod-down 68.0%

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

       9. pow-flip 68.0%

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

       10. metadata-eval 68.0%

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

   16. Applied egg-rr 68.0%

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

       1. cube-prod 83.1%

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

   18. Simplified 83.1%

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

   if 8e12 < k

   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 58.6%

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

      1. associate-*l/ 58.6%

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

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in t around 0 71.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 8000000000000:\\ \;\;\;\;\frac{2}{\left(k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 12000000000000:\\ \;\;\;\;\frac{2}{\left(k \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t\_m}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \end{array} \end{array} \]

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 8.2e-93)
    (/ 2.0 (/ (* 2.0 (/ (pow (* t_m (pow (cbrt k) 2.0)) 3.0) l)) l))
    (if (<= k 12000000000000.0)
      (/ 2.0 (* (* k (pow (* t_m (pow (cbrt l) -2.0)) 3.0)) (* 2.0 k)))
      (/
       2.0
       (/ (* (pow k 2.0) (* (/ t_m l) (/ (pow (sin k) 2.0) (cos k)))) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e-93) {
		tmp = 2.0 / ((2.0 * (pow((t_m * pow(cbrt(k), 2.0)), 3.0) / l)) / l);
	} else if (k <= 12000000000000.0) {
		tmp = 2.0 / ((k * pow((t_m * pow(cbrt(l), -2.0)), 3.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * ((t_m / l) * (pow(sin(k), 2.0) / cos(k)))) / 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 tmp;
	if (k <= 8.2e-93) {
		tmp = 2.0 / ((2.0 * (Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0) / l)) / l);
	} else if (k <= 12000000000000.0) {
		tmp = 2.0 / ((k * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * ((t_m / l) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)))) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8.2e-93)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0) / l)) / l));
	elseif (k <= 12000000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(Float64(t_m / l) * Float64((sin(k) ^ 2.0) / cos(k)))) / 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_] := N[(t$95$s * If[LessEqual[k, 8.2e-93], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 12000000000000.0], N[(2.0 / N[(N[(k * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 8.1999999999999998e-93

   1. Initial program 56.6%

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

   3. Simplified 58.2%

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

      1. associate-*l/ 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in k around 0 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow3 54.4%

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

      3. *-commutative 54.4%

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

      4. cbrt-prod 54.4%

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

      5. rem-cbrt-cube 63.6%

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

      6. unpow2 63.6%

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

      7. cbrt-prod 71.5%

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

      8. pow2 71.5%

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

   9. Applied egg-rr 71.5%

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

   if 8.1999999999999998e-93 < k < 1.2e13

   1. Initial program 51.8%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in k around 0 68.7%

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

      1. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.5%

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

      3. add-sqr-sqrt 34.1%

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

      4. associate-*l* 34.2%

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

      5. sqrt-div 26.3%

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

      6. sqrt-pow1 26.3%

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

      7. metadata-eval 26.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      8. sqrt-div 26.3%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      9. sqrt-pow1 26.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      10. metadata-eval 26.2%

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

   9. Applied egg-rr 26.2%

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

       1. associate-*r/ 26.2%

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

       2. *-rgt-identity 26.2%

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

       3. associate-/l/ 26.2%

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

   11. Simplified 26.2%

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

   12. Taylor expanded in k around 0 68.7%

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

       1. associate-/l* 68.7%

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

   14. Simplified 68.7%

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

       1. rem-cube-cbrt 68.5%

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

       2. cbrt-div 68.5%

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

       3. rem-cbrt-cube 69.3%

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

       4. unpow2 69.3%

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

       5. cbrt-prod 83.0%

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

       6. unpow2 83.0%

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

       7. div-inv 83.0%

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

       8. unpow-prod-down 68.0%

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

       9. pow-flip 68.0%

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

       10. metadata-eval 68.0%

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

   16. Applied egg-rr 68.0%

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

       1. cube-prod 83.1%

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

   18. Simplified 83.1%

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

   if 1.2e13 < k

   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 58.6%

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

      1. associate-*l/ 58.6%

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

   6. Applied egg-rr 58.6%

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

   7. Taylor expanded in t around 0 71.4%

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

      1. associate-/l* 71.3%

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

      2. times-frac 71.3%

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

   9. Simplified 71.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 12000000000000:\\ \;\;\;\;\frac{2}{\left(k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{1}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 7.2e+246)
    (/ 2.0 (/ (* 2.0 (/ (pow (* t_m (pow (cbrt k) 2.0)) 3.0) l)) l))
    (/
     2.0
     (* (* 2.0 k) (* (sin k) (/ 1.0 (pow (/ (pow (cbrt l) 2.0) t_m) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.2e+246) {
		tmp = 2.0 / ((2.0 * (pow((t_m * pow(cbrt(k), 2.0)), 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * (1.0 / pow((pow(cbrt(l), 2.0) / t_m), 3.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.2e+246) {
		tmp = 2.0 / ((2.0 * (Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * (1.0 / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 7.2e+246)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64(1.0 / (Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 7.2e+246], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.2e246

   1. Initial program 56.7%

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

   3. Simplified 58.9%

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

      1. associate-*l/ 59.6%

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

   6. Applied egg-rr 59.6%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. add-cube-cbrt 57.1%

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

      2. pow3 57.1%

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

      3. *-commutative 57.1%

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

      4. cbrt-prod 57.1%

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

      5. rem-cbrt-cube 64.9%

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

      6. unpow2 64.9%

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

      7. cbrt-prod 70.8%

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

      8. pow2 70.8%

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

   9. Applied egg-rr 70.8%

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

   if 7.2e246 < l

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 58.1%

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

      1. *-commutative 58.1%

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

   7. Simplified 58.1%

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

      1. unpow3 58.1%

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

      2. *-un-lft-identity 58.1%

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

      3. times-frac 58.1%

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

      4. pow2 58.1%

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

      5. pow2 58.1%

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

   9. Applied egg-rr 58.1%

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

       1. /-rgt-identity 58.1%

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

       2. associate-*r/ 58.1%

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

       3. unpow2 58.1%

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

       4. unpow3 58.1%

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

       5. rem-cube-cbrt 58.1%

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

       6. cbrt-div 58.1%

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

       7. rem-cbrt-cube 59.3%

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

       8. unpow2 59.3%

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

       9. cbrt-prod 79.0%

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

       10. unpow2 79.0%

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

       11. clear-num 78.9%

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

       12. cube-div 79.0%

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

       13. metadata-eval 79.0%

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

   11. Applied egg-rr 79.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{1}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+247}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.15e+247)
    (/ 2.0 (/ (* 2.0 (/ (pow (* t_m (pow (cbrt k) 2.0)) 3.0) l)) l))
    (/ 2.0 (* (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)) (* 2.0 k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.15e+247) {
		tmp = 2.0 / ((2.0 * (pow((t_m * pow(cbrt(k), 2.0)), 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.15e+247) {
		tmp = 2.0 / ((2.0 * (Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.15e+247)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.15e+247], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.14999999999999995e247

   1. Initial program 56.7%

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

   3. Simplified 58.9%

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

      1. associate-*l/ 59.6%

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

   6. Applied egg-rr 59.6%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. add-cube-cbrt 57.1%

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

      2. pow3 57.1%

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

      3. *-commutative 57.1%

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

      4. cbrt-prod 57.1%

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

      5. rem-cbrt-cube 64.9%

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

      6. unpow2 64.9%

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

      7. cbrt-prod 70.8%

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

      8. pow2 70.8%

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

   9. Applied egg-rr 70.8%

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

   if 1.14999999999999995e247 < l

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 58.1%

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

      1. *-commutative 58.1%

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

   7. Simplified 58.1%

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

      1. associate-/r* 44.3%

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

      2. add-cube-cbrt 44.3%

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

      3. associate-*l* 44.3%

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

      4. associate-/r* 43.5%

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

      5. cbrt-div 43.5%

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

      6. rem-cbrt-cube 43.5%

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

      7. cbrt-prod 44.3%

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

      8. pow2 44.3%

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

      9. pow2 44.3%

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

      10. associate-/r* 43.5%

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

      11. cbrt-div 43.5%

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

      12. rem-cbrt-cube 44.7%

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

      13. cbrt-prod 71.1%

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

      14. pow2 71.1%

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

   9. Applied egg-rr 79.2%

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

       1. unpow2 71.1%

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

       2. cube-mult 70.8%

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

   11. Simplified 79.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.15 \cdot 10^{+247}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 5.4e+246)
    (/ 2.0 (/ (* 2.0 (/ (pow (* t_m (pow (cbrt k) 2.0)) 3.0) l)) l))
    (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5.4e+246) {
		tmp = 2.0 / ((2.0 * (pow((t_m * pow(cbrt(k), 2.0)), 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 5.4e+246) {
		tmp = 2.0 / ((2.0 * (Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 5.4e+246)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 5.4e+246], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 5.4e246

   1. Initial program 56.7%

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

   3. Simplified 58.9%

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

      1. associate-*l/ 59.6%

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

   6. Applied egg-rr 59.6%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. add-cube-cbrt 57.1%

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

      2. pow3 57.1%

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

      3. *-commutative 57.1%

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

      4. cbrt-prod 57.1%

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

      5. rem-cbrt-cube 64.9%

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

      6. unpow2 64.9%

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

      7. cbrt-prod 70.8%

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

      8. pow2 70.8%

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

   9. Applied egg-rr 70.8%

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

   if 5.4e246 < l

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 58.1%

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

      1. *-commutative 58.1%

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

   7. Simplified 58.1%

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

      1. associate-/r* 44.3%

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

      2. add-sqr-sqrt 22.4%

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

      3. pow2 22.4%

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

      4. associate-/r* 22.1%

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

      5. sqrt-div 22.1%

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

      6. sqrt-pow1 22.2%

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

      7. metadata-eval 22.2%

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

      8. sqrt-prod 29.0%

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

      9. add-sqr-sqrt 29.0%

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

   9. Applied egg-rr 22.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4.9e+246)
    (/ 2.0 (/ (* 2.0 (/ (pow (* k (pow t_m 1.5)) 2.0) l)) l))
    (/ 2.0 (* (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)) (* 2.0 k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.9e+246) {
		tmp = 2.0 / ((2.0 * (pow((k * pow(t_m, 1.5)), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((sin(k) * pow((pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.9d+246) then
        tmp = 2.0d0 / ((2.0d0 * (((k * (t_m ** 1.5d0)) ** 2.0d0) / l)) / l)
    else
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)) * (2.0d0 * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4.9e+246) {
		tmp = 2.0 / ((2.0 * (Math.pow((k * Math.pow(t_m, 1.5)), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 4.9e+246:
		tmp = 2.0 / ((2.0 * (math.pow((k * math.pow(t_m, 1.5)), 2.0) / l)) / l)
	else:
		tmp = 2.0 / ((math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)) * (2.0 * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 4.9e+246)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(k * (t_m ^ 1.5)) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 4.9e+246)
		tmp = 2.0 / ((2.0 * (((k * (t_m ^ 1.5)) ^ 2.0) / l)) / l);
	else
		tmp = 2.0 / ((sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)) * (2.0 * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 4.9e+246], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.90000000000000028e246

   1. Initial program 56.7%

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

   3. Simplified 58.9%

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

      1. associate-*l/ 59.6%

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

   6. Applied egg-rr 59.6%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. pow1 57.2%

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

      2. add-sqr-sqrt 24.4%

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

      3. pow2 24.4%

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

      4. sqrt-prod 24.4%

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

      5. sqrt-pow1 27.0%

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

      6. metadata-eval 27.0%

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

      7. pow1 27.0%

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

      8. sqrt-pow1 29.3%

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

      9. metadata-eval 29.3%

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

   9. Applied egg-rr 29.3%

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

       1. unpow1 29.3%

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

   11. Simplified 29.3%

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

   if 4.90000000000000028e246 < l

   1. Initial program 43.5%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in k around 0 58.1%

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

      1. *-commutative 58.1%

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

   7. Simplified 58.1%

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

      1. associate-/r* 44.3%

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

      2. add-sqr-sqrt 22.4%

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

      3. pow2 22.4%

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

      4. associate-/r* 22.1%

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

      5. sqrt-div 22.1%

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

      6. sqrt-pow1 22.2%

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

      7. metadata-eval 22.2%

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

      8. sqrt-prod 29.0%

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

      9. add-sqr-sqrt 29.0%

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

   9. Applied egg-rr 22.4%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t}^{1.5}\right)}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.1e-92)
    (/ 2.0 (/ (* 2.0 (/ (pow (* k (pow t_m 1.5)) 2.0) l)) l))
    (/ 2.0 (* (* k (pow (* t_m (pow (cbrt l) -2.0)) 3.0)) (* 2.0 k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.1e-92) {
		tmp = 2.0 / ((2.0 * (pow((k * pow(t_m, 1.5)), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((k * pow((t_m * pow(cbrt(l), -2.0)), 3.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.1e-92) {
		tmp = 2.0 / ((2.0 * (Math.pow((k * Math.pow(t_m, 1.5)), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((k * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0)) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.1e-92)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(k * (t_m ^ 1.5)) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(k * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.1e-92], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.09999999999999994e-92

   1. Initial program 56.6%

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

   3. Simplified 58.2%

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

      1. associate-*l/ 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in k around 0 54.4%

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

      1. pow1 54.4%

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

      2. add-sqr-sqrt 21.5%

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

      3. pow2 21.5%

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

      4. sqrt-prod 21.5%

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

      5. sqrt-pow1 25.0%

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

      6. metadata-eval 25.0%

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

      7. pow1 25.0%

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

      8. sqrt-pow1 27.0%

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

      9. metadata-eval 27.0%

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

   9. Applied egg-rr 27.0%

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

       1. unpow1 27.0%

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

   11. Simplified 27.0%

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

   if 1.09999999999999994e-92 < k

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in k around 0 57.4%

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

      1. *-commutative 57.4%

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

   7. Simplified 57.4%

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

      1. associate-/r* 59.1%

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

      2. div-inv 59.1%

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

      3. add-sqr-sqrt 31.7%

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

      4. associate-*l* 31.8%

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

      5. sqrt-div 19.2%

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

      6. sqrt-pow1 19.2%

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

      7. metadata-eval 19.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      8. sqrt-div 19.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      9. sqrt-pow1 19.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      10. metadata-eval 19.1%

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

   9. Applied egg-rr 19.1%

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

       1. associate-*r/ 19.1%

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

       2. *-rgt-identity 19.1%

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

       3. associate-/l/ 20.3%

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

   11. Simplified 20.3%

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

   12. Taylor expanded in k around 0 58.3%

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

       1. associate-/l* 58.3%

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

   14. Simplified 58.3%

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

       1. rem-cube-cbrt 58.3%

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

       2. cbrt-div 58.3%

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

       3. rem-cbrt-cube 61.4%

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

       4. unpow2 61.4%

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

       5. cbrt-prod 66.5%

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

       6. unpow2 66.5%

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

       7. div-inv 66.5%

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

       8. unpow-prod-down 58.1%

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

       9. pow-flip 58.1%

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

       10. metadata-eval 58.1%

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

   16. Applied egg-rr 58.1%

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

       1. cube-prod 66.5%

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

   18. Simplified 66.5%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t}^{1.5}\right)}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot {\left(\frac{\sqrt{t\_m}}{\sqrt[3]{\ell}}\right)}^{6}\right)}\\ \end{array} \end{array} \]

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 (<= k 2.2e-92)
    (/ 2.0 (/ (* 2.0 (/ (pow (* k (pow t_m 1.5)) 2.0) l)) l))
    (/ 2.0 (* (* 2.0 k) (* k (pow (/ (sqrt t_m) (cbrt l)) 6.0)))))))

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 <= 2.2e-92) {
		tmp = 2.0 / ((2.0 * (pow((k * pow(t_m, 1.5)), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (k * pow((sqrt(t_m) / cbrt(l)), 6.0)));
	}
	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 <= 2.2e-92) {
		tmp = 2.0 / ((2.0 * (Math.pow((k * Math.pow(t_m, 1.5)), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (k * Math.pow((Math.sqrt(t_m) / Math.cbrt(l)), 6.0)));
	}
	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 <= 2.2e-92)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64((Float64(k * (t_m ^ 1.5)) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k * (Float64(sqrt(t_m) / cbrt(l)) ^ 6.0))));
	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, 2.2e-92], N[(2.0 / N[(N[(2.0 * N[(N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k * N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.19999999999999987e-92

   1. Initial program 56.6%

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

   3. Simplified 58.2%

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

      1. associate-*l/ 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in k around 0 54.4%

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

      1. pow1 54.4%

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

      2. add-sqr-sqrt 21.5%

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

      3. pow2 21.5%

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

      4. sqrt-prod 21.5%

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

      5. sqrt-pow1 25.0%

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

      6. metadata-eval 25.0%

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

      7. pow1 25.0%

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

      8. sqrt-pow1 27.0%

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

      9. metadata-eval 27.0%

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

   9. Applied egg-rr 27.0%

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

       1. unpow1 27.0%

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

   11. Simplified 27.0%

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

   if 2.19999999999999987e-92 < k

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in k around 0 57.4%

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

      1. *-commutative 57.4%

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

   7. Simplified 57.4%

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

      1. associate-/r* 59.1%

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

      2. div-inv 59.1%

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

      3. add-sqr-sqrt 31.7%

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

      4. associate-*l* 31.8%

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

      5. sqrt-div 19.2%

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

      6. sqrt-pow1 19.2%

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

      7. metadata-eval 19.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      8. sqrt-div 19.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      9. sqrt-pow1 19.1%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

      10. metadata-eval 19.1%

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

   9. Applied egg-rr 19.1%

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

       1. associate-*r/ 19.1%

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

       2. *-rgt-identity 19.1%

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

       3. associate-/l/ 20.3%

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

   11. Simplified 20.3%

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

   12. Taylor expanded in k around 0 58.3%

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

       1. associate-/l* 58.3%

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

   14. Simplified 58.3%

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

       1. rem-cube-cbrt 58.3%

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

       2. cbrt-div 58.3%

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

       3. rem-cbrt-cube 61.4%

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

       4. unpow2 61.4%

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

       5. cbrt-prod 66.5%

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

       6. unpow2 66.5%

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

       7. add-sqr-sqrt 34.7%

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

       8. unpow-prod-down 34.7%

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

       9. sqrt-div 33.3%

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

       10. sqrt-pow1 26.8%

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

       11. metadata-eval 26.8%

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

       12. pow1 26.8%

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

       13. sqrt-div 26.8%

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

       14. sqrt-pow1 33.3%

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

       15. metadata-eval 33.3%

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

       16. pow1 33.3%

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

   16. Applied egg-rr 33.3%

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

       1. pow-sqr 33.3%

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

       2. metadata-eval 33.3%

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

   18. Simplified 33.3%

       \[\leadsto \frac{2}{\left(k \cdot \color{blue}{{\left(\frac{\sqrt{t}}{\sqrt[3]{\ell}}\right)}^{6}}\right) \cdot \left(k \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t}^{1.5}\right)}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot {\left(\frac{\sqrt{t}}{\sqrt[3]{\ell}}\right)}^{6}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{2 \cdot \frac{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}}{\ell}}{\ell}} \end{array} \]

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 (pow t_m 1.5)) 2.0) l)) l))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * (pow((k * pow(t_m, 1.5)), 2.0) / l)) / 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 * (t_m ** 1.5d0)) ** 2.0d0) / l)) / 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 * Math.pow(t_m, 1.5)), 2.0) / l)) / 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 * math.pow(t_m, 1.5)), 2.0) / l)) / 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 * Float64((Float64(k * (t_m ^ 1.5)) ^ 2.0) / l)) / 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 * (t_m ^ 1.5)) ^ 2.0) / l)) / 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[(N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 58.1%

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

   1. associate-*l/ 59.2%

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

6. Applied egg-rr 59.2%

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

7. Taylor expanded in k around 0 56.4%

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

   1. pow1 56.4%

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

   2. add-sqr-sqrt 23.8%

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

   3. pow2 23.8%

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

   4. sqrt-prod 23.8%

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

   5. sqrt-pow1 26.7%

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

   6. metadata-eval 26.7%

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

   7. pow1 26.7%

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

   8. sqrt-pow1 28.5%

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

   9. metadata-eval 28.5%

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

9. Applied egg-rr 28.5%

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

    1. unpow1 28.5%

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

11. Simplified 28.5%

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

Alternative 14: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)} \end{array} \]

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 k) (* k (pow (/ (pow t_m 1.5) 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) {
	return t_s * (2.0 / ((2.0 * k) * (k * pow((pow(t_m, 1.5) / l), 2.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * k) * (k * (((t_m ** 1.5d0) / l) ** 2.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * k) * (k * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * k) * (k * math.pow((math.pow(t_m, 1.5) / l), 2.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k * (Float64((t_m ^ 1.5) / l) ^ 2.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * k) * (k * (((t_m ^ 1.5) / l) ^ 2.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 56.0%

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

5. Taylor expanded in k around 0 55.1%

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

   1. *-commutative 55.1%

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

7. Simplified 55.1%

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

   1. associate-/r* 59.4%

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

   2. div-inv 59.4%

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

   3. add-sqr-sqrt 31.7%

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

   4. associate-*l* 31.7%

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

   5. sqrt-div 13.6%

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

   6. sqrt-pow1 13.5%

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

   7. metadata-eval 13.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   8. sqrt-div 13.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   9. sqrt-pow1 15.1%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. metadata-eval 15.1%

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

9. Applied egg-rr 15.1%

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

    1. associate-*r/ 15.1%

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

    2. *-rgt-identity 15.1%

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

    3. associate-/l/ 14.3%

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

11. Simplified 14.3%

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

12. Taylor expanded in k around 0 56.5%

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

    1. associate-/l* 56.2%

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

14. Simplified 56.2%

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

    1. add-sqr-sqrt 24.5%

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

    2. sqrt-div 24.5%

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

    3. sqrt-pow1 23.3%

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

    4. metadata-eval 23.3%

       \[\leadsto \frac{2}{\left(k \cdot \left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{{\ell}^{2}}} \cdot \sqrt{\frac{{t}^{3}}{{\ell}^{2}}}\right)\right) \cdot \left(k \cdot 2\right)} \]

    5. sqrt-pow1 17.4%

       \[\leadsto \frac{2}{\left(k \cdot \left(\frac{{t}^{1.5}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{{t}^{3}}{{\ell}^{2}}}\right)\right) \cdot \left(k \cdot 2\right)} \]

    6. metadata-eval 17.4%

       \[\leadsto \frac{2}{\left(k \cdot \left(\frac{{t}^{1.5}}{{\ell}^{\color{blue}{1}}} \cdot \sqrt{\frac{{t}^{3}}{{\ell}^{2}}}\right)\right) \cdot \left(k \cdot 2\right)} \]

    7. pow1 17.4%

       \[\leadsto \frac{2}{\left(k \cdot \left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\frac{{t}^{3}}{{\ell}^{2}}}\right)\right) \cdot \left(k \cdot 2\right)} \]

    8. sqrt-div 17.4%

       \[\leadsto \frac{2}{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{{\ell}^{2}}}}\right)\right) \cdot \left(k \cdot 2\right)} \]

    9. sqrt-pow1 19.4%

       \[\leadsto \frac{2}{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{{\ell}^{2}}}\right)\right) \cdot \left(k \cdot 2\right)} \]

    10. metadata-eval 19.4%

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

    11. sqrt-pow1 28.2%

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

    12. metadata-eval 28.2%

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

    13. pow1 28.2%

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

16. Applied egg-rr 28.2%

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

    1. unpow2 28.2%

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

18. Simplified 28.2%

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

19. Final simplification 28.2%

    \[\leadsto \frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \]
20. Add Preprocessing

Alternative 15: 53.3% accurate, 2.0× speedup

Math

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

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 k) (* k (/ (pow t_m 3.0) (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) {
	return t_s * (2.0 / ((2.0 * k) * (k * (pow(t_m, 3.0) / pow(l, 2.0)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * k) * (k * ((t_m ** 3.0d0) / (l ** 2.0d0)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * k) * (k * (Math.pow(t_m, 3.0) / Math.pow(l, 2.0)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * k) * (k * (math.pow(t_m, 3.0) / math.pow(l, 2.0)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k * Float64((t_m ^ 3.0) / (l ^ 2.0))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * k) * (k * ((t_m ^ 3.0) / (l ^ 2.0)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 56.0%

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

5. Taylor expanded in k around 0 55.1%

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

   1. *-commutative 55.1%

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

7. Simplified 55.1%

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

   1. associate-/r* 59.4%

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

   2. div-inv 59.4%

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

   3. add-sqr-sqrt 31.7%

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

   4. associate-*l* 31.7%

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

   5. sqrt-div 13.6%

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

   6. sqrt-pow1 13.5%

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

   7. metadata-eval 13.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   8. sqrt-div 13.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   9. sqrt-pow1 15.1%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. metadata-eval 15.1%

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

9. Applied egg-rr 15.1%

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

    1. associate-*r/ 15.1%

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

    2. *-rgt-identity 15.1%

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

    3. associate-/l/ 14.3%

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

11. Simplified 14.3%

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

12. Taylor expanded in k around 0 56.5%

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

    1. associate-/l* 56.2%

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

14. Simplified 56.2%

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

15. Final simplification 56.2%

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

Alternative 16: 53.2% accurate, 2.0× speedup

Math

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

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 (/ 2.0 (* (* 2.0 k) (* k (* (pow t_m 3.0) (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) {
	return t_s * (2.0 / ((2.0 * k) * (k * (pow(t_m, 3.0) * pow(l, -2.0)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((2.0d0 * k) * (k * ((t_m ** 3.0d0) * (l ** (-2.0d0))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * k) * (k * (Math.pow(t_m, 3.0) * Math.pow(l, -2.0)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((2.0 * k) * (k * (math.pow(t_m, 3.0) * math.pow(l, -2.0)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k * Float64((t_m ^ 3.0) * (l ^ -2.0))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((2.0 * k) * (k * ((t_m ^ 3.0) * (l ^ -2.0)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 56.0%

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

5. Taylor expanded in k around 0 55.1%

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

   1. *-commutative 55.1%

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

7. Simplified 55.1%

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

   1. associate-/r* 59.4%

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

   2. div-inv 59.4%

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

   3. add-sqr-sqrt 31.7%

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

   4. associate-*l* 31.7%

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

   5. sqrt-div 13.6%

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

   6. sqrt-pow1 13.5%

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

   7. metadata-eval 13.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell}} \cdot \left(\sqrt{\frac{{t}^{3}}{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   8. sqrt-div 13.5%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{{t}^{3}}}{\sqrt{\ell}}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   9. sqrt-pow1 15.1%

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\sqrt{\ell}} \cdot \left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell}} \cdot \frac{1}{\ell}\right)\right) \cdot \sin k\right) \cdot \left(k \cdot 2\right)} \]

   10. metadata-eval 15.1%

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

9. Applied egg-rr 15.1%

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

    1. associate-*r/ 15.1%

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

    2. *-rgt-identity 15.1%

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

    3. associate-/l/ 14.3%

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

11. Simplified 14.3%

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

12. Taylor expanded in k around 0 56.5%

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

    1. associate-/l* 56.2%

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

14. Simplified 56.2%

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

    1. div-inv 56.1%

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

    2. pow-flip 56.1%

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

    3. metadata-eval 56.1%

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

16. Applied egg-rr 56.1%

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

17. Final simplification 56.1%

    \[\leadsto \frac{2}{\left(2 \cdot k\right) \cdot \left(k \cdot \left({t}^{3} \cdot {\ell}^{-2}\right)\right)} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))))