Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.5% → 86.4%

Time: 28.0s

Alternatives: 20

Speedup: 2.0×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% 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: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{2}{\sin k}}\\ t_3 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t\_2}{t\_3}\right)}^{2}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{t\_2}{t\_3 \cdot \tan k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (/ 2.0 (sin k)))) (t_3 (/ t_m (pow (cbrt l) 2.0))))
   (*
    t_s
    (if (<= t_m 1.3e-99)
      (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
      (*
       (/ (pow (/ t_2 t_3) 2.0) (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ t_2 (* t_3 (tan k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((2.0 / sin(k)));
	double t_3 = t_m / pow(cbrt(l), 2.0);
	double tmp;
	if (t_m <= 1.3e-99) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else {
		tmp = (pow((t_2 / t_3), 2.0) / (2.0 + pow((k / t_m), 2.0))) * (t_2 / (t_3 * tan(k)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((2.0 / Math.sin(k)));
	double t_3 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t_m <= 1.3e-99) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else {
		tmp = (Math.pow((t_2 / t_3), 2.0) / (2.0 + Math.pow((k / t_m), 2.0))) * (t_2 / (t_3 * Math.tan(k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(2.0 / sin(k)))
	t_3 = Float64(t_m / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.3e-99)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64((Float64(t_2 / t_3) ^ 2.0) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(t_2 / Float64(t_3 * tan(k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-99], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Power[N[(t$95$2 / t$95$3), $MachinePrecision], 2.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(t$95$3 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.30000000000000003e-99

   1. Initial program 46.1%

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

   3. Simplified 52.4%

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

      1. add-sqr-sqrt 36.2%

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

      2. pow2 36.2%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in k around inf 34.4%

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

   if 1.30000000000000003e-99 < t

   1. Initial program 64.9%

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

   3. Simplified 73.2%

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

      1. associate-/l/ 73.2%

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

      2. add-cube-cbrt 73.0%

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

      3. times-frac 73.0%

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

   6. Applied egg-rr 93.1%

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

      1. associate-/l/ 93.1%

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

   8. Simplified 93.1%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t_3 := \sqrt[3]{\frac{1}{k}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-102}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{\frac{2}{t\_2} \cdot \frac{\frac{1}{\sin k}}{t\_2}}{\tan k}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot \left(t\_3 \cdot t\_3\right)\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)) (t_3 (cbrt (/ 1.0 k))))
   (*
    t_s
    (if (<= t_m 1.5e-102)
      (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
      (if (<= t_m 3.3e+192)
        (/
         (/ (* (/ 2.0 t_2) (/ (/ 1.0 (sin k)) t_2)) (tan k))
         (+ 2.0 (pow (/ k t_m) 2.0)))
        (pow (* (/ (pow (cbrt l) 2.0) t_m) (* t_3 t_3)) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(t_m, 1.5) / l;
	double t_3 = cbrt((1.0 / k));
	double tmp;
	if (t_m <= 1.5e-102) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 3.3e+192) {
		tmp = (((2.0 / t_2) * ((1.0 / sin(k)) / t_2)) / tan(k)) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * (t_3 * t_3)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(t_m, 1.5) / l;
	double t_3 = Math.cbrt((1.0 / k));
	double tmp;
	if (t_m <= 1.5e-102) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 3.3e+192) {
		tmp = (((2.0 / t_2) * ((1.0 / Math.sin(k)) / t_2)) / Math.tan(k)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * (t_3 * t_3)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((t_m ^ 1.5) / l)
	t_3 = cbrt(Float64(1.0 / k))
	tmp = 0.0
	if (t_m <= 1.5e-102)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 3.3e+192)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_2) * Float64(Float64(1.0 / sin(k)) / t_2)) / tan(k)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * Float64(t_3 * t_3)) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(1.0 / k), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.5e-102], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 3.3e+192], N[(N[(N[(N[(2.0 / t$95$2), $MachinePrecision] * N[(N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.5e-102

   1. Initial program 46.1%

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

   3. Simplified 52.4%

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

      1. add-sqr-sqrt 36.2%

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

      2. pow2 36.2%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in k around inf 34.4%

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

   if 1.5e-102 < t < 3.3000000000000001e192

   1. Initial program 62.6%

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

   3. Simplified 69.1%

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

      1. div-inv 69.1%

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

      2. add-sqr-sqrt 69.1%

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

      3. times-frac 69.1%

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

      4. associate-/r* 62.5%

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

      5. sqrt-div 62.5%

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

      6. sqrt-pow1 62.5%

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

      7. metadata-eval 62.5%

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

      8. sqrt-prod 41.8%

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

      9. add-sqr-sqrt 59.7%

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

      10. associate-/r* 54.6%

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

      11. sqrt-div 54.7%

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

      12. sqrt-pow1 58.0%

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

      13. metadata-eval 58.0%

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

      14. sqrt-prod 52.4%

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

      15. add-sqr-sqrt 87.4%

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

   6. Applied egg-rr 87.4%

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

   if 3.3000000000000001e192 < t

   1. Initial program 71.0%

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

   3. Simplified 84.0%

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

      1. associate-/l/ 84.0%

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

      2. add-cube-cbrt 84.0%

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

      3. times-frac 84.0%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in k around 0 58.5%

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

       1. *-commutative 58.5%

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

       2. associate-/r* 58.5%

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

   11. Simplified 58.5%

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

       1. add-cube-cbrt 58.5%

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

       2. pow2 58.5%

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

       3. div-inv 58.5%

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

       4. cbrt-prod 58.5%

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

       5. cbrt-div 58.5%

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

       6. unpow2 58.5%

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

       7. cbrt-prod 58.5%

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

       8. unpow2 58.5%

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

       9. unpow3 58.5%

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

       10. add-cbrt-cube 58.5%

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

       11. pow-flip 58.5%

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

       12. metadata-eval 58.5%

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

       13. div-inv 58.5%

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

   13. Applied egg-rr 79.4%

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

       1. unpow2 79.4%

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

       2. unpow3 79.4%

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

   15. Simplified 79.4%

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

       1. pow1/3 78.8%

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

       2. sqr-pow 78.8%

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

       3. unpow-prod-down 41.9%

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

       4. metadata-eval 41.9%

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

       5. unpow-1 41.9%

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

       6. metadata-eval 41.9%

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

       7. unpow-1 41.9%

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

   17. Applied egg-rr 41.9%

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

       1. unpow1/3 41.9%

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

       2. unpow1/3 95.4%

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

   19. Simplified 95.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-102}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{{t}^{1.5}}{\ell}} \cdot \frac{\frac{1}{\sin k}}{\frac{{t}^{1.5}}{\ell}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \left(\sqrt[3]{\frac{1}{k}} \cdot \sqrt[3]{\frac{1}{k}}\right)\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{k}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-103}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\ell \cdot {\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t\_m}\right)}^{3}}{\tan k}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot \left(t\_2 \cdot t\_2\right)\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (/ 1.0 k))))
   (*
    t_s
    (if (<= t_m 4.2e-103)
      (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
      (if (<= t_m 6.2e+171)
        (/
         (/ (* l (pow (/ (cbrt (* l (/ 2.0 (sin k)))) t_m) 3.0)) (tan k))
         (+ 2.0 (pow (/ k t_m) 2.0)))
        (pow (* (/ (pow (cbrt l) 2.0) t_m) (* t_2 t_2)) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((1.0 / k));
	double tmp;
	if (t_m <= 4.2e-103) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 6.2e+171) {
		tmp = ((l * pow((cbrt((l * (2.0 / sin(k)))) / t_m), 3.0)) / tan(k)) / (2.0 + pow((k / t_m), 2.0));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * (t_2 * t_2)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((1.0 / k));
	double tmp;
	if (t_m <= 4.2e-103) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 6.2e+171) {
		tmp = ((l * Math.pow((Math.cbrt((l * (2.0 / Math.sin(k)))) / t_m), 3.0)) / Math.tan(k)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * (t_2 * t_2)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(1.0 / k))
	tmp = 0.0
	if (t_m <= 4.2e-103)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 6.2e+171)
		tmp = Float64(Float64(Float64(l * (Float64(cbrt(Float64(l * Float64(2.0 / sin(k)))) / t_m) ^ 3.0)) / tan(k)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * Float64(t_2 * t_2)) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(1.0 / k), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-103], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 6.2e+171], N[(N[(N[(l * N[Power[N[(N[Power[N[(l * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.20000000000000009e-103

   1. Initial program 46.1%

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

   3. Simplified 52.4%

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

      1. add-sqr-sqrt 36.2%

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

      2. pow2 36.2%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in k around inf 34.4%

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

   if 4.20000000000000009e-103 < t < 6.1999999999999998e171

   1. Initial program 66.7%

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

   3. Simplified 73.9%

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

      1. associate-/r/ 75.6%

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

      2. div-inv 75.5%

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

      3. clear-num 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. add-cube-cbrt 75.2%

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

      2. pow3 75.2%

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

      3. *-commutative 75.2%

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

      4. cbrt-prod 75.2%

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

      5. associate-*r/ 75.2%

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

      6. cbrt-div 75.0%

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

      7. unpow3 75.0%

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

      8. add-cbrt-cube 85.3%

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

   8. Applied egg-rr 85.3%

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

      1. cube-prod 85.3%

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

      2. rem-cube-cbrt 85.5%

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

      3. *-commutative 85.5%

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

   10. Simplified 85.5%

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

   if 6.1999999999999998e171 < t

   1. Initial program 61.6%

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

   3. Simplified 72.0%

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

      1. associate-/l/ 72.0%

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

      2. add-cube-cbrt 72.0%

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

      3. times-frac 72.0%

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

   6. Applied egg-rr 99.0%

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

      1. associate-/l/ 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in k around 0 51.7%

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

       1. *-commutative 51.7%

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

       2. associate-/r* 51.7%

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

   11. Simplified 51.7%

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

       1. add-cube-cbrt 51.7%

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

       2. pow2 51.7%

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

       3. div-inv 51.7%

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

       4. cbrt-prod 51.7%

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

       5. cbrt-div 51.7%

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

       6. unpow2 51.7%

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

       7. cbrt-prod 51.7%

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

       8. unpow2 51.7%

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

       9. unpow3 51.7%

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

       10. add-cbrt-cube 51.7%

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

       11. pow-flip 51.7%

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

       12. metadata-eval 51.7%

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

       13. div-inv 51.7%

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

   13. Applied egg-rr 71.7%

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

       1. unpow2 71.7%

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

       2. unpow3 71.7%

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

   15. Simplified 71.7%

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

       1. pow1/3 70.9%

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

       2. sqr-pow 70.9%

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

       3. unpow-prod-down 35.8%

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

       4. metadata-eval 35.8%

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

       5. unpow-1 35.8%

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

       6. metadata-eval 35.8%

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

       7. unpow-1 35.8%

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

   17. Applied egg-rr 35.8%

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

       1. unpow1/3 35.8%

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

       2. unpow1/3 92.8%

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

   19. Simplified 92.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-103}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\ell \cdot {\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t}\right)}^{3}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \left(\sqrt[3]{\frac{1}{k}} \cdot \sqrt[3]{\frac{1}{k}}\right)\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{k}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-99}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot \left(t\_2 \cdot t\_2\right)\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (/ 1.0 k))))
   (*
    t_s
    (if (<= t_m 2.9e-99)
      (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
      (if (<= t_m 8.6e+110)
        (*
         (/
          (* (/ 2.0 (sin k)) (/ l (pow t_m 3.0)))
          (+ 2.0 (pow (/ k t_m) 2.0)))
         (/ l (tan k)))
        (pow (* (/ (pow (cbrt l) 2.0) t_m) (* t_2 t_2)) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((1.0 / k));
	double tmp;
	if (t_m <= 2.9e-99) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / sin(k)) * (l / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) * (l / tan(k));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * (t_2 * t_2)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((1.0 / k));
	double tmp;
	if (t_m <= 2.9e-99) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / Math.sin(k)) * (l / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * (t_2 * t_2)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(1.0 / k))
	tmp = 0.0
	if (t_m <= 2.9e-99)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 8.6e+110)
		tmp = Float64(Float64(Float64(Float64(2.0 / sin(k)) * Float64(l / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * Float64(t_2 * t_2)) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(1.0 / k), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-99], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 8.6e+110], N[(N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.89999999999999985e-99

   1. Initial program 46.1%

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

   3. Simplified 52.4%

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

      1. add-sqr-sqrt 36.2%

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

      2. pow2 36.2%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in k around inf 34.4%

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

   if 2.89999999999999985e-99 < t < 8.60000000000000014e110

   1. Initial program 72.0%

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

   3. Simplified 81.2%

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

      1. associate-/l/ 81.2%

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

      2. associate-/r/ 83.6%

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

      3. times-frac 85.8%

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

      4. div-inv 85.8%

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

      5. clear-num 85.8%

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

   6. Applied egg-rr 85.8%

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

   if 8.60000000000000014e110 < t

   1. Initial program 58.3%

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

   3. Simplified 65.7%

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

      1. associate-/l/ 65.7%

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

      2. add-cube-cbrt 65.7%

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

      3. times-frac 65.7%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/l/ 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in k around 0 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-/r* 49.1%

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

   11. Simplified 49.1%

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

       1. add-cube-cbrt 49.1%

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

       2. pow2 49.1%

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

       3. div-inv 49.1%

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

       4. cbrt-prod 49.1%

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

       5. cbrt-div 49.1%

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

       6. unpow2 49.1%

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

       7. cbrt-prod 49.1%

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

       8. unpow2 49.1%

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

       9. unpow3 49.1%

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

       10. add-cbrt-cube 49.1%

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

       11. pow-flip 49.1%

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

       12. metadata-eval 49.1%

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

       13. div-inv 49.1%

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

   13. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. unpow3 65.5%

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

   15. Simplified 65.5%

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

       1. pow1/3 64.7%

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

       2. sqr-pow 64.7%

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

       3. unpow-prod-down 33.3%

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

       4. metadata-eval 33.3%

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

       5. unpow-1 33.3%

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

       6. metadata-eval 33.3%

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

       7. unpow-1 33.3%

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

   17. Applied egg-rr 33.3%

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

       1. unpow1/3 33.4%

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

       2. unpow1/3 86.4%

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

   19. Simplified 86.4%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-99}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \left(\sqrt[3]{\frac{1}{k}} \cdot \sqrt[3]{\frac{1}{k}}\right)\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-113)
    (/ 2.0 (* (* k k) (/ (* t_m (pow (sin k) 2.0)) (* (cos k) (pow l 2.0)))))
    (if (<= t_m 8.6e+110)
      (*
       (/ (* (/ 2.0 (sin k)) (/ l (pow t_m 3.0))) (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ l (tan k)))
      (pow
       (*
        (/ (pow (cbrt l) 2.0) t_m)
        (exp (* (* -2.0 (log k)) 0.3333333333333333)))
       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-113) {
		tmp = 2.0 / ((k * k) * ((t_m * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))));
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / sin(k)) * (l / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) * (l / tan(k));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * exp(((-2.0 * log(k)) * 0.3333333333333333))), 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-113) {
		tmp = 2.0 / ((k * k) * ((t_m * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0))));
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / Math.sin(k)) * (l / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.exp(((-2.0 * Math.log(k)) * 0.3333333333333333))), 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-113)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0)))));
	elseif (t_m <= 8.6e+110)
		tmp = Float64(Float64(Float64(Float64(2.0 / sin(k)) * Float64(l / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * exp(Float64(Float64(-2.0 * log(k)) * 0.3333333333333333))) ^ 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-113], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e+110], N[(N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Exp[N[(N[(-2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.1200000000000001e-113

   1. Initial program 46.1%

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

      1. unpow3 46.0%

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

      2. times-frac 59.5%

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

      3. pow2 59.5%

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

   4. Applied egg-rr 59.5%

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

   5. Taylor expanded in t around 0 53.1%

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

      1. associate-/l* 53.6%

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

      2. *-commutative 53.6%

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

   7. Simplified 53.6%

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

      1. unpow2 42.3%

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

   9. Applied egg-rr 53.6%

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

   if 1.1200000000000001e-113 < t < 8.60000000000000014e110

   1. Initial program 71.0%

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

   3. Simplified 79.8%

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

      1. associate-/l/ 79.8%

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

      2. associate-/r/ 82.0%

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

      3. times-frac 84.2%

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

      4. div-inv 84.1%

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

      5. clear-num 84.2%

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

   6. Applied egg-rr 84.2%

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

   if 8.60000000000000014e110 < t

   1. Initial program 58.3%

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

   3. Simplified 65.7%

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

      1. associate-/l/ 65.7%

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

      2. add-cube-cbrt 65.7%

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

      3. times-frac 65.7%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/l/ 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in k around 0 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-/r* 49.1%

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

   11. Simplified 49.1%

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

       1. add-cube-cbrt 49.1%

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

       2. pow2 49.1%

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

       3. div-inv 49.1%

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

       4. cbrt-prod 49.1%

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

       5. cbrt-div 49.1%

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

       6. unpow2 49.1%

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

       7. cbrt-prod 49.1%

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

       8. unpow2 49.1%

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

       9. unpow3 49.1%

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

       10. add-cbrt-cube 49.1%

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

       11. pow-flip 49.1%

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

       12. metadata-eval 49.1%

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

       13. div-inv 49.1%

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

   13. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. unpow3 65.5%

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

   15. Simplified 65.5%

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

       1. pow1/3 64.7%

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

       2. pow-to-exp 64.9%

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

       3. log-pow 33.3%

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

   17. Applied egg-rr 33.3%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-99}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t\_m}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.15e-99)
    (pow (* (sqrt (/ (cos k) t_m)) (* (/ l k) (/ (sqrt 2.0) (sin k)))) 2.0)
    (if (<= t_m 8.6e+110)
      (*
       (/ (* (/ 2.0 (sin k)) (/ l (pow t_m 3.0))) (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ l (tan k)))
      (pow
       (*
        (/ (pow (cbrt l) 2.0) t_m)
        (exp (* (* -2.0 (log k)) 0.3333333333333333)))
       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 <= 1.15e-99) {
		tmp = pow((sqrt((cos(k) / t_m)) * ((l / k) * (sqrt(2.0) / sin(k)))), 2.0);
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / sin(k)) * (l / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) * (l / tan(k));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * exp(((-2.0 * log(k)) * 0.3333333333333333))), 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 <= 1.15e-99) {
		tmp = Math.pow((Math.sqrt((Math.cos(k) / t_m)) * ((l / k) * (Math.sqrt(2.0) / Math.sin(k)))), 2.0);
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / Math.sin(k)) * (l / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.exp(((-2.0 * Math.log(k)) * 0.3333333333333333))), 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 <= 1.15e-99)
		tmp = Float64(sqrt(Float64(cos(k) / t_m)) * Float64(Float64(l / k) * Float64(sqrt(2.0) / sin(k)))) ^ 2.0;
	elseif (t_m <= 8.6e+110)
		tmp = Float64(Float64(Float64(Float64(2.0 / sin(k)) * Float64(l / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * exp(Float64(Float64(-2.0 * log(k)) * 0.3333333333333333))) ^ 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, 1.15e-99], N[Power[N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 8.6e+110], N[(N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Exp[N[(N[(-2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.1499999999999999e-99

   1. Initial program 46.1%

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

   3. Simplified 52.4%

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

      1. add-sqr-sqrt 36.2%

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

      2. pow2 36.2%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in k around inf 34.4%

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

      1. times-frac 34.4%

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

   9. Simplified 34.4%

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

   if 1.1499999999999999e-99 < t < 8.60000000000000014e110

   1. Initial program 72.0%

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

   3. Simplified 81.2%

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

      1. associate-/l/ 81.2%

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

      2. associate-/r/ 83.6%

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

      3. times-frac 85.8%

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

      4. div-inv 85.8%

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

      5. clear-num 85.8%

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

   6. Applied egg-rr 85.8%

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

   if 8.60000000000000014e110 < t

   1. Initial program 58.3%

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

   3. Simplified 65.7%

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

      1. associate-/l/ 65.7%

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

      2. add-cube-cbrt 65.7%

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

      3. times-frac 65.7%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/l/ 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in k around 0 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-/r* 49.1%

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

   11. Simplified 49.1%

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

       1. add-cube-cbrt 49.1%

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

       2. pow2 49.1%

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

       3. div-inv 49.1%

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

       4. cbrt-prod 49.1%

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

       5. cbrt-div 49.1%

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

       6. unpow2 49.1%

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

       7. cbrt-prod 49.1%

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

       8. unpow2 49.1%

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

       9. unpow3 49.1%

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

       10. add-cbrt-cube 49.1%

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

       11. pow-flip 49.1%

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

       12. metadata-eval 49.1%

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

       13. div-inv 49.1%

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

   13. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. unpow3 65.5%

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

   15. Simplified 65.5%

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

       1. pow1/3 64.7%

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

       2. pow-to-exp 64.9%

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

       3. log-pow 33.3%

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

   17. Applied egg-rr 33.3%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-99}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right)\right)}^{2}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-98}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e-98)
    (pow (* (/ (* l (sqrt 2.0)) (* k (sin k))) (sqrt (/ (cos k) t_m))) 2.0)
    (if (<= t_m 8.6e+110)
      (*
       (/ (* (/ 2.0 (sin k)) (/ l (pow t_m 3.0))) (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ l (tan k)))
      (pow
       (*
        (/ (pow (cbrt l) 2.0) t_m)
        (exp (* (* -2.0 (log k)) 0.3333333333333333)))
       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 <= 2.1e-98) {
		tmp = pow((((l * sqrt(2.0)) / (k * sin(k))) * sqrt((cos(k) / t_m))), 2.0);
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / sin(k)) * (l / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) * (l / tan(k));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * exp(((-2.0 * log(k)) * 0.3333333333333333))), 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 <= 2.1e-98) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k * Math.sin(k))) * Math.sqrt((Math.cos(k) / t_m))), 2.0);
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / Math.sin(k)) * (l / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.exp(((-2.0 * Math.log(k)) * 0.3333333333333333))), 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 <= 2.1e-98)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k * sin(k))) * sqrt(Float64(cos(k) / t_m))) ^ 2.0;
	elseif (t_m <= 8.6e+110)
		tmp = Float64(Float64(Float64(Float64(2.0 / sin(k)) * Float64(l / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * exp(Float64(Float64(-2.0 * log(k)) * 0.3333333333333333))) ^ 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, 2.1e-98], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 8.6e+110], N[(N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Exp[N[(N[(-2.0 * N[Log[k], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.09999999999999992e-98

   1. Initial program 46.1%

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

   3. Simplified 52.4%

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

      1. add-sqr-sqrt 36.2%

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

      2. pow2 36.2%

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

   6. Applied egg-rr 5.9%

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

   7. Taylor expanded in k around inf 34.4%

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

   if 2.09999999999999992e-98 < t < 8.60000000000000014e110

   1. Initial program 72.0%

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

   3. Simplified 81.2%

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

      1. associate-/l/ 81.2%

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

      2. associate-/r/ 83.6%

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

      3. times-frac 85.8%

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

      4. div-inv 85.8%

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

      5. clear-num 85.8%

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

   6. Applied egg-rr 85.8%

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

   if 8.60000000000000014e110 < t

   1. Initial program 58.3%

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

   3. Simplified 65.7%

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

      1. associate-/l/ 65.7%

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

      2. add-cube-cbrt 65.7%

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

      3. times-frac 65.7%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/l/ 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in k around 0 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-/r* 49.1%

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

   11. Simplified 49.1%

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

       1. add-cube-cbrt 49.1%

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

       2. pow2 49.1%

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

       3. div-inv 49.1%

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

       4. cbrt-prod 49.1%

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

       5. cbrt-div 49.1%

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

       6. unpow2 49.1%

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

       7. cbrt-prod 49.1%

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

       8. unpow2 49.1%

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

       9. unpow3 49.1%

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

       10. add-cbrt-cube 49.1%

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

       11. pow-flip 49.1%

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

       12. metadata-eval 49.1%

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

       13. div-inv 49.1%

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

   13. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. unpow3 65.5%

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

   15. Simplified 65.5%

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

       1. pow1/3 64.7%

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

       2. pow-to-exp 64.9%

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

       3. log-pow 33.3%

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

   17. Applied egg-rr 33.3%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-98}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot e^{\left(-2 \cdot \log k\right) \cdot 0.3333333333333333}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k}}{{t\_m}^{3}}\right) \cdot \frac{\ell}{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-113)
    (/ 2.0 (* (* k k) (/ (* t_m (pow (sin k) 2.0)) (* (cos k) (pow l 2.0)))))
    (if (<= t_m 8.6e+110)
      (*
       (* l (/ (/ 2.0 (sin k)) (pow t_m 3.0)))
       (/ l (* (+ 2.0 (pow (/ k t_m) 2.0)) (tan k))))
      (pow (* (/ (pow (cbrt l) 2.0) t_m) (pow k -0.6666666666666666)) 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-113) {
		tmp = 2.0 / ((k * k) * ((t_m * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))));
	} else if (t_m <= 8.6e+110) {
		tmp = (l * ((2.0 / sin(k)) / pow(t_m, 3.0))) * (l / ((2.0 + pow((k / t_m), 2.0)) * tan(k)));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * pow(k, -0.6666666666666666)), 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-113) {
		tmp = 2.0 / ((k * k) * ((t_m * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0))));
	} else if (t_m <= 8.6e+110) {
		tmp = (l * ((2.0 / Math.sin(k)) / Math.pow(t_m, 3.0))) * (l / ((2.0 + Math.pow((k / t_m), 2.0)) * Math.tan(k)));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.pow(k, -0.6666666666666666)), 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-113)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0)))));
	elseif (t_m <= 8.6e+110)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / sin(k)) / (t_m ^ 3.0))) * Float64(l / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * tan(k))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * (k ^ -0.6666666666666666)) ^ 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-113], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e+110], N[(N[(l * N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.1200000000000001e-113

   1. Initial program 46.1%

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

      1. unpow3 46.0%

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

      2. times-frac 59.5%

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

      3. pow2 59.5%

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

   4. Applied egg-rr 59.5%

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

   5. Taylor expanded in t around 0 53.1%

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

      1. associate-/l* 53.6%

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

      2. *-commutative 53.6%

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

   7. Simplified 53.6%

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

      1. unpow2 42.3%

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

   9. Applied egg-rr 53.6%

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

   if 1.1200000000000001e-113 < t < 8.60000000000000014e110

   1. Initial program 71.0%

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

   3. Simplified 79.8%

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

      1. add-cube-cbrt 79.5%

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

      2. pow3 79.4%

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

      3. cbrt-div 79.3%

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

      4. rem-cbrt-cube 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. associate-/l/ 81.6%

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

      3. associate-/r/ 83.6%

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

      4. cube-div 81.6%

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

      5. pow3 81.5%

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

      6. add-cube-cbrt 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-/l* 84.3%

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

      3. associate-/r/ 79.9%

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

   10. Simplified 79.9%

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

   if 8.60000000000000014e110 < t

   1. Initial program 58.3%

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

   3. Simplified 65.7%

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

      1. associate-/l/ 65.7%

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

      2. add-cube-cbrt 65.7%

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

      3. times-frac 65.7%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/l/ 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in k around 0 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-/r* 49.1%

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

   11. Simplified 49.1%

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

       1. add-cube-cbrt 49.1%

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

       2. pow2 49.1%

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

       3. div-inv 49.1%

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

       4. cbrt-prod 49.1%

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

       5. cbrt-div 49.1%

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

       6. unpow2 49.1%

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

       7. cbrt-prod 49.1%

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

       8. unpow2 49.1%

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

       9. unpow3 49.1%

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

       10. add-cbrt-cube 49.1%

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

       11. pow-flip 49.1%

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

       12. metadata-eval 49.1%

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

       13. div-inv 49.1%

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

   13. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. unpow3 65.5%

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

   15. Simplified 65.5%

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

       1. pow1/3 64.7%

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

       2. pow-pow 33.3%

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

       3. metadata-eval 33.3%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \]

   17. Applied egg-rr 33.3%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{{k}^{-0.6666666666666666}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{\sin k}}{{t}^{3}}\right) \cdot \frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-113)
    (/ 2.0 (* (* k k) (/ (* t_m (pow (sin k) 2.0)) (* (cos k) (pow l 2.0)))))
    (if (<= t_m 8.6e+110)
      (*
       (/ (* (/ 2.0 (sin k)) (/ l (pow t_m 3.0))) (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ l (tan k)))
      (pow (* (/ (pow (cbrt l) 2.0) t_m) (pow k -0.6666666666666666)) 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-113) {
		tmp = 2.0 / ((k * k) * ((t_m * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))));
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / sin(k)) * (l / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) * (l / tan(k));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * pow(k, -0.6666666666666666)), 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-113) {
		tmp = 2.0 / ((k * k) * ((t_m * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0))));
	} else if (t_m <= 8.6e+110) {
		tmp = (((2.0 / Math.sin(k)) * (l / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.pow(k, -0.6666666666666666)), 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-113)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0)))));
	elseif (t_m <= 8.6e+110)
		tmp = Float64(Float64(Float64(Float64(2.0 / sin(k)) * Float64(l / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * (k ^ -0.6666666666666666)) ^ 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-113], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.6e+110], N[(N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.1200000000000001e-113

   1. Initial program 46.1%

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

      1. unpow3 46.0%

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

      2. times-frac 59.5%

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

      3. pow2 59.5%

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

   4. Applied egg-rr 59.5%

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

   5. Taylor expanded in t around 0 53.1%

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

      1. associate-/l* 53.6%

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

      2. *-commutative 53.6%

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

   7. Simplified 53.6%

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

      1. unpow2 42.3%

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

   9. Applied egg-rr 53.6%

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

   if 1.1200000000000001e-113 < t < 8.60000000000000014e110

   1. Initial program 71.0%

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

   3. Simplified 79.8%

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

      1. associate-/l/ 79.8%

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

      2. associate-/r/ 82.0%

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

      3. times-frac 84.2%

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

      4. div-inv 84.1%

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

      5. clear-num 84.2%

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

   6. Applied egg-rr 84.2%

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

   if 8.60000000000000014e110 < t

   1. Initial program 58.3%

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

   3. Simplified 65.7%

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

      1. associate-/l/ 65.7%

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

      2. add-cube-cbrt 65.7%

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

      3. times-frac 65.7%

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

   6. Applied egg-rr 98.8%

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

      1. associate-/l/ 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in k around 0 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-/r* 49.1%

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

   11. Simplified 49.1%

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

       1. add-cube-cbrt 49.1%

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

       2. pow2 49.1%

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

       3. div-inv 49.1%

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

       4. cbrt-prod 49.1%

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

       5. cbrt-div 49.1%

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

       6. unpow2 49.1%

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

       7. cbrt-prod 49.1%

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

       8. unpow2 49.1%

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

       9. unpow3 49.1%

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

       10. add-cbrt-cube 49.1%

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

       11. pow-flip 49.1%

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

       12. metadata-eval 49.1%

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

       13. div-inv 49.1%

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

   13. Applied egg-rr 65.5%

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

       1. unpow2 65.5%

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

       2. unpow3 65.5%

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

   15. Simplified 65.5%

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

       1. pow1/3 64.7%

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

       2. pow-pow 33.3%

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

       3. metadata-eval 33.3%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \]

   17. Applied egg-rr 33.3%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{{k}^{-0.6666666666666666}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.4e-22)
    (pow (* (/ (pow (cbrt l) 2.0) t_m) (pow k -0.6666666666666666)) 3.0)
    (/
     2.0
     (* (* k k) (/ (* t_m (pow (sin k) 2.0)) (* (cos k) (pow l 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-22) {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * pow(k, -0.6666666666666666)), 3.0);
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * pow(sin(k), 2.0)) / (cos(k) * pow(l, 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-22) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.pow(k, -0.6666666666666666)), 3.0);
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * Math.pow(Math.sin(k), 2.0)) / (Math.cos(k) * Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.4e-22)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * (k ^ -0.6666666666666666)) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / Float64(cos(k) * (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.4e-22], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000002e-22

   1. Initial program 52.5%

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

   3. Simplified 60.6%

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

      1. associate-/l/ 60.6%

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

      2. add-cube-cbrt 60.4%

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

      3. times-frac 60.4%

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

   6. Applied egg-rr 83.9%

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

      1. associate-/l/ 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in k around 0 43.5%

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

       1. *-commutative 43.5%

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

       2. associate-/r* 44.2%

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

   11. Simplified 44.2%

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

       1. add-cube-cbrt 44.2%

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

       2. pow2 44.2%

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

       3. div-inv 42.5%

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

       4. cbrt-prod 42.5%

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

       5. cbrt-div 42.5%

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

       6. unpow2 42.5%

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

       7. cbrt-prod 42.5%

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

       8. unpow2 42.5%

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

       9. unpow3 42.5%

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

       10. add-cbrt-cube 42.5%

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

       11. pow-flip 42.5%

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

       12. metadata-eval 42.5%

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

       13. div-inv 42.5%

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

   13. Applied egg-rr 60.2%

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

       1. unpow2 60.2%

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

       2. unpow3 60.2%

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

   15. Simplified 60.2%

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

       1. pow1/3 59.3%

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

       2. pow-pow 26.1%

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

       3. metadata-eval 26.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \]

   17. Applied egg-rr 26.1%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{{k}^{-0.6666666666666666}}\right)}^{3} \]

   if 2.40000000000000002e-22 < k

   1. Initial program 52.5%

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

      1. unpow3 52.5%

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

      2. times-frac 60.8%

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

      3. pow2 60.8%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in t around 0 66.5%

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

      1. associate-/l* 66.5%

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

      2. *-commutative 66.5%

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

   7. Simplified 66.5%

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

      1. unpow2 49.6%

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

   9. Applied egg-rr 66.5%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t\_m}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.4e-22)
    (pow (* (/ (pow (cbrt l) 2.0) t_m) (pow k -0.6666666666666666)) 3.0)
    (/ 2.0 (* (pow k 2.0) (* (/ (pow k 2.0) (pow l 2.0)) (/ t_m (cos k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-22) {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * pow(k, -0.6666666666666666)), 3.0);
	} else {
		tmp = 2.0 / (pow(k, 2.0) * ((pow(k, 2.0) / pow(l, 2.0)) * (t_m / cos(k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-22) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.pow(k, -0.6666666666666666)), 3.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((Math.pow(k, 2.0) / Math.pow(l, 2.0)) * (t_m / Math.cos(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.4e-22)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * (k ^ -0.6666666666666666)) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64((k ^ 2.0) / (l ^ 2.0)) * Float64(t_m / cos(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.4e-22], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000002e-22

   1. Initial program 52.5%

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

   3. Simplified 60.6%

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

      1. associate-/l/ 60.6%

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

      2. add-cube-cbrt 60.4%

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

      3. times-frac 60.4%

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

   6. Applied egg-rr 83.9%

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

      1. associate-/l/ 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in k around 0 43.5%

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

       1. *-commutative 43.5%

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

       2. associate-/r* 44.2%

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

   11. Simplified 44.2%

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

       1. add-cube-cbrt 44.2%

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

       2. pow2 44.2%

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

       3. div-inv 42.5%

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

       4. cbrt-prod 42.5%

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

       5. cbrt-div 42.5%

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

       6. unpow2 42.5%

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

       7. cbrt-prod 42.5%

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

       8. unpow2 42.5%

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

       9. unpow3 42.5%

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

       10. add-cbrt-cube 42.5%

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

       11. pow-flip 42.5%

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

       12. metadata-eval 42.5%

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

       13. div-inv 42.5%

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

   13. Applied egg-rr 60.2%

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

       1. unpow2 60.2%

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

       2. unpow3 60.2%

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

   15. Simplified 60.2%

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

       1. pow1/3 59.3%

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

       2. pow-pow 26.1%

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

       3. metadata-eval 26.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \]

   17. Applied egg-rr 26.1%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{{k}^{-0.6666666666666666}}\right)}^{3} \]

   if 2.40000000000000002e-22 < k

   1. Initial program 52.5%

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

      1. unpow3 52.5%

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

      2. times-frac 60.8%

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

      3. pow2 60.8%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in t around 0 66.5%

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

      1. associate-/l* 66.5%

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

      2. *-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in k around 0 63.8%

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

      1. times-frac 63.9%

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

   10. Applied egg-rr 63.9%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-67)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (if (<= t_m 3.2e-17)
      (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (pow k 2.0))
      (/ (pow l 2.0) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-67) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else if (t_m <= 3.2e-17) {
		tmp = pow((pow(cbrt(l), 2.0) / t_m), 3.0) / pow(k, 2.0);
	} else {
		tmp = pow(l, 2.0) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-67) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else if (t_m <= 3.2e-17) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / Math.pow(k, 2.0);
	} else {
		tmp = Math.pow(l, 2.0) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-67)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	elseif (t_m <= 3.2e-17)
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / (k ^ 2.0));
	else
		tmp = Float64((l ^ 2.0) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-67], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e-17], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.8000000000000001e-67

   1. Initial program 47.5%

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

      1. unpow3 47.5%

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

      2. times-frac 61.1%

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

      3. pow2 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in t around 0 54.1%

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

      1. associate-/l* 54.5%

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

      2. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in k around 0 49.9%

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

      1. associate-/l* 50.8%

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

   10. Simplified 50.8%

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

   if 2.8000000000000001e-67 < t < 3.2000000000000002e-17

   1. Initial program 67.5%

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

   3. Simplified 75.6%

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

      1. associate-/l/ 75.6%

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

      2. add-cube-cbrt 75.5%

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

      3. times-frac 75.5%

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

   6. Applied egg-rr 82.8%

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

      1. associate-/l/ 82.8%

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

   8. Simplified 82.8%

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

   9. Taylor expanded in k around 0 43.1%

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

       1. *-commutative 43.1%

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

       2. associate-/r* 58.4%

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

   11. Simplified 58.4%

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

       1. add-cube-cbrt 58.1%

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

       2. pow2 58.1%

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

       3. cbrt-div 58.2%

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

       4. unpow2 58.2%

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

       5. cbrt-prod 58.1%

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

       6. unpow2 58.1%

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

       7. unpow3 58.1%

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

       8. add-cbrt-cube 58.1%

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

       9. cbrt-div 58.4%

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

       10. unpow2 58.4%

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

       11. cbrt-prod 66.4%

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

       12. unpow2 66.4%

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

       13. unpow3 66.4%

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

       14. add-cbrt-cube 66.4%

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

   13. Applied egg-rr 66.4%

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

       1. unpow2 66.4%

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

       2. unpow3 66.4%

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

   15. Simplified 66.4%

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

   if 3.2000000000000002e-17 < t

   1. Initial program 63.5%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in k around 0 50.8%

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

      1. *-commutative 50.8%

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

   7. Simplified 50.8%

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

      1. add-cube-cbrt 50.8%

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

      2. pow2 50.8%

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

      3. cbrt-div 50.8%

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

      4. cbrt-prod 50.7%

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

      5. unpow3 50.7%

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

      6. add-cbrt-cube 50.8%

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

      7. unpow2 50.8%

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

      8. cbrt-prod 50.8%

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

      9. pow2 50.8%

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

      10. cbrt-div 50.8%

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

      11. cbrt-prod 50.8%

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

      12. unpow3 50.8%

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

      13. add-cbrt-cube 54.3%

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

      14. unpow2 54.3%

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

      15. cbrt-prod 69.8%

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

      16. pow2 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. unpow2 69.8%

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

       2. unpow3 69.7%

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

       3. cube-div 68.0%

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

       4. rem-cube-cbrt 68.0%

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

   11. Simplified 68.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-24)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (/ (pow l 2.0) (pow (* t_m (pow (cbrt k) 2.0)) 3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-24) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = pow(l, 2.0) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-24) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = Math.pow(l, 2.0) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-24)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64((l ^ 2.0) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-24], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000002e-24

   1. Initial program 48.5%

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

      1. unpow3 48.4%

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

      2. times-frac 61.8%

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

      3. pow2 61.8%

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

   4. Applied egg-rr 61.8%

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

   5. Taylor expanded in t around 0 53.6%

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

      1. associate-/l* 54.1%

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

      2. *-commutative 54.1%

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

   7. Simplified 54.1%

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

   8. Taylor expanded in k around 0 49.2%

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

      1. associate-/l* 50.1%

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

   10. Simplified 50.1%

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

   if 2.8000000000000002e-24 < t

   1. Initial program 64.1%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in k around 0 51.6%

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

      1. *-commutative 51.6%

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

   7. Simplified 51.6%

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

      1. add-cube-cbrt 51.6%

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

      2. pow2 51.6%

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

      3. cbrt-div 51.5%

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

      4. cbrt-prod 51.5%

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

      5. unpow3 51.5%

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

      6. add-cbrt-cube 51.5%

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

      7. unpow2 51.5%

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

      8. cbrt-prod 51.5%

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

      9. pow2 51.5%

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

      10. cbrt-div 51.6%

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

      11. cbrt-prod 51.5%

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

      12. unpow3 51.5%

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

      13. add-cbrt-cube 55.0%

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

      14. unpow2 55.0%

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

      15. cbrt-prod 70.2%

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

      16. pow2 70.2%

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

   9. Applied egg-rr 70.2%

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

       1. unpow2 70.2%

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

       2. unpow3 70.2%

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

       3. cube-div 68.4%

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

       4. rem-cube-cbrt 68.4%

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

   11. Simplified 68.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.4e-22)
    (pow (* (/ (pow (cbrt l) 2.0) t_m) (pow k -0.6666666666666666)) 3.0)
    (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-22) {
		tmp = pow(((pow(cbrt(l), 2.0) / t_m) * pow(k, -0.6666666666666666)), 3.0);
	} else {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.4e-22) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / t_m) * Math.pow(k, -0.6666666666666666)), 3.0);
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.4e-22)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / t_m) * (k ^ -0.6666666666666666)) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.4e-22], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k, -0.6666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000002e-22

   1. Initial program 52.5%

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

   3. Simplified 60.6%

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

      1. associate-/l/ 60.6%

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

      2. add-cube-cbrt 60.4%

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

      3. times-frac 60.4%

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

   6. Applied egg-rr 83.9%

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

      1. associate-/l/ 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in k around 0 43.5%

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

       1. *-commutative 43.5%

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

       2. associate-/r* 44.2%

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

   11. Simplified 44.2%

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

       1. add-cube-cbrt 44.2%

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

       2. pow2 44.2%

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

       3. div-inv 42.5%

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

       4. cbrt-prod 42.5%

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

       5. cbrt-div 42.5%

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

       6. unpow2 42.5%

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

       7. cbrt-prod 42.5%

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

       8. unpow2 42.5%

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

       9. unpow3 42.5%

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

       10. add-cbrt-cube 42.5%

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

       11. pow-flip 42.5%

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

       12. metadata-eval 42.5%

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

       13. div-inv 42.5%

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

   13. Applied egg-rr 60.2%

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

       1. unpow2 60.2%

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

       2. unpow3 60.2%

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

   15. Simplified 60.2%

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

       1. pow1/3 59.3%

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

       2. pow-pow 26.1%

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

       3. metadata-eval 26.1%

          \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{\color{blue}{-0.6666666666666666}}\right)}^{3} \]

   17. Applied egg-rr 26.1%

       \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \color{blue}{{k}^{-0.6666666666666666}}\right)}^{3} \]

   if 2.40000000000000002e-22 < k

   1. Initial program 52.5%

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

      1. unpow3 52.5%

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

      2. times-frac 60.8%

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

      3. pow2 60.8%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in t around 0 66.5%

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

      1. associate-/l* 66.5%

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

      2. *-commutative 66.5%

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

   7. Simplified 66.5%

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

   8. Taylor expanded in k around 0 63.8%

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

   9. Taylor expanded in k around inf 59.2%

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

       1. associate-/l* 59.3%

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

   11. Simplified 59.3%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot {k}^{-0.6666666666666666}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t\_m}^{3}}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-67)
    (/ 2.0 (* (pow k 2.0) (* (pow k 2.0) (/ t_m (pow l 2.0)))))
    (/ (/ (pow l 2.0) (pow t_m 3.0)) (* k k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-67) {
		tmp = 2.0 / (pow(k, 2.0) * (pow(k, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = (pow(l, 2.0) / pow(t_m, 3.0)) / (k * 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 (t_m <= 2.6d-67) then
        tmp = 2.0d0 / ((k ** 2.0d0) * ((k ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = ((l ** 2.0d0) / (t_m ** 3.0d0)) / (k * 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 (t_m <= 2.6e-67) {
		tmp = 2.0 / (Math.pow(k, 2.0) * (Math.pow(k, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(t_m, 3.0)) / (k * 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 t_m <= 2.6e-67:
		tmp = 2.0 / (math.pow(k, 2.0) * (math.pow(k, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(t_m, 3.0)) / (k * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.6e-67)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64((k ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (t_m ^ 3.0)) / Float64(k * 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 (t_m <= 2.6e-67)
		tmp = 2.0 / ((k ^ 2.0) * ((k ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = ((l ^ 2.0) / (t_m ^ 3.0)) / (k * 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[t$95$m, 2.6e-67], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5999999999999999e-67

   1. Initial program 47.5%

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

      1. unpow3 47.5%

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

      2. times-frac 61.1%

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

      3. pow2 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in t around 0 54.1%

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

      1. associate-/l* 54.5%

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

      2. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in k around 0 49.9%

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

      1. associate-/l* 50.8%

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

   10. Simplified 50.8%

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

   if 2.5999999999999999e-67 < t

   1. Initial program 64.2%

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

   3. Simplified 71.1%

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

      1. associate-/l/ 71.1%

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

      2. add-cube-cbrt 70.9%

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

      3. times-frac 70.9%

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

   6. Applied egg-rr 93.7%

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

      1. associate-/l/ 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in k around 0 49.6%

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

       1. *-commutative 49.6%

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

       2. associate-/r* 52.0%

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

   11. Simplified 52.0%

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

       1. unpow2 52.0%

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

   13. Applied egg-rr 52.0%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t\_m}^{3}}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-67)
    (/ 2.0 (* (pow k 4.0) (/ t_m (* (cos k) (pow l 2.0)))))
    (/ (/ (pow l 2.0) (pow t_m 3.0)) (* k k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-67) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / (cos(k) * pow(l, 2.0))));
	} else {
		tmp = (pow(l, 2.0) / pow(t_m, 3.0)) / (k * 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 (t_m <= 2.8d-67) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (cos(k) * (l ** 2.0d0))))
    else
        tmp = ((l ** 2.0d0) / (t_m ** 3.0d0)) / (k * 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 (t_m <= 2.8e-67) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / (Math.cos(k) * Math.pow(l, 2.0))));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(t_m, 3.0)) / (k * 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 t_m <= 2.8e-67:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / (math.cos(k) * math.pow(l, 2.0))))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(t_m, 3.0)) / (k * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-67)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / Float64(cos(k) * (l ^ 2.0)))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (t_m ^ 3.0)) / Float64(k * 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 (t_m <= 2.8e-67)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (cos(k) * (l ^ 2.0))));
	else
		tmp = ((l ^ 2.0) / (t_m ^ 3.0)) / (k * 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[t$95$m, 2.8e-67], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-67

   1. Initial program 47.5%

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

      1. unpow3 47.5%

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

      2. times-frac 61.1%

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

      3. pow2 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in t around 0 54.1%

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

      1. associate-/l* 54.5%

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

      2. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in k around 0 52.4%

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

   9. Taylor expanded in k around inf 49.1%

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

       1. associate-/l* 50.4%

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

   11. Simplified 50.4%

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

   if 2.8000000000000001e-67 < t

   1. Initial program 64.2%

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

   3. Simplified 71.1%

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

      1. associate-/l/ 71.1%

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

      2. add-cube-cbrt 70.9%

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

      3. times-frac 70.9%

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

   6. Applied egg-rr 93.7%

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

      1. associate-/l/ 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in k around 0 49.6%

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

       1. *-commutative 49.6%

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

       2. associate-/r* 52.0%

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

   11. Simplified 52.0%

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

       1. unpow2 52.0%

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

   13. Applied egg-rr 52.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{1}{\frac{{\ell}^{2}}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t\_m}^{3}}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-67)
    (/ 2.0 (* (pow k 4.0) (/ 1.0 (/ (pow l 2.0) t_m))))
    (/ (/ (pow l 2.0) (pow t_m 3.0)) (* k k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-67) {
		tmp = 2.0 / (pow(k, 4.0) * (1.0 / (pow(l, 2.0) / t_m)));
	} else {
		tmp = (pow(l, 2.0) / pow(t_m, 3.0)) / (k * 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 (t_m <= 2.6d-67) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (1.0d0 / ((l ** 2.0d0) / t_m)))
    else
        tmp = ((l ** 2.0d0) / (t_m ** 3.0d0)) / (k * 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 (t_m <= 2.6e-67) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (1.0 / (Math.pow(l, 2.0) / t_m)));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(t_m, 3.0)) / (k * 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 t_m <= 2.6e-67:
		tmp = 2.0 / (math.pow(k, 4.0) * (1.0 / (math.pow(l, 2.0) / t_m)))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(t_m, 3.0)) / (k * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.6e-67)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(1.0 / Float64((l ^ 2.0) / t_m))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (t_m ^ 3.0)) / Float64(k * 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 (t_m <= 2.6e-67)
		tmp = 2.0 / ((k ^ 4.0) * (1.0 / ((l ^ 2.0) / t_m)));
	else
		tmp = ((l ^ 2.0) / (t_m ^ 3.0)) / (k * 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[t$95$m, 2.6e-67], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(1.0 / N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.5999999999999999e-67

   1. Initial program 47.5%

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

      1. unpow3 47.5%

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

      2. times-frac 61.1%

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

      3. pow2 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in t around 0 54.1%

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

      1. associate-/l* 54.5%

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

      2. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in k around 0 47.8%

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

      1. associate-/l* 49.1%

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

   10. Simplified 49.1%

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

       1. clear-num 49.1%

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

       2. inv-pow 49.1%

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

   12. Applied egg-rr 49.1%

       \[\leadsto \frac{2}{{k}^{4} \cdot \color{blue}{{\left(\frac{{\ell}^{2}}{t}\right)}^{-1}}} \]
   13. Step-by-step derivation

       1. unpow-1 49.1%

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

   14. Simplified 49.1%

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

   if 2.5999999999999999e-67 < t

   1. Initial program 64.2%

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

   3. Simplified 71.1%

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

      1. associate-/l/ 71.1%

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

      2. add-cube-cbrt 70.9%

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

      3. times-frac 70.9%

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

   6. Applied egg-rr 93.7%

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

      1. associate-/l/ 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in k around 0 49.6%

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

       1. *-commutative 49.6%

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

       2. associate-/r* 52.0%

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

   11. Simplified 52.0%

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

       1. unpow2 52.0%

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

   13. Applied egg-rr 52.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{1}{\frac{{\ell}^{2}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t\_m}^{3}}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-67)
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0))))
    (/ (/ (pow l 2.0) (pow t_m 3.0)) (* k k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-67) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(l, 2.0)));
	} else {
		tmp = (pow(l, 2.0) / pow(t_m, 3.0)) / (k * 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 (t_m <= 2.8d-67) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else
        tmp = ((l ** 2.0d0) / (t_m ** 3.0d0)) / (k * 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 (t_m <= 2.8e-67) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(t_m, 3.0)) / (k * 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 t_m <= 2.8e-67:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(l, 2.0)))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(t_m, 3.0)) / (k * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-67)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	else
		tmp = Float64(Float64((l ^ 2.0) / (t_m ^ 3.0)) / Float64(k * 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 (t_m <= 2.8e-67)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (l ^ 2.0)));
	else
		tmp = ((l ^ 2.0) / (t_m ^ 3.0)) / (k * 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[t$95$m, 2.8e-67], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-67

   1. Initial program 47.5%

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

      1. unpow3 47.5%

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

      2. times-frac 61.1%

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

      3. pow2 61.1%

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

   4. Applied egg-rr 61.1%

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

   5. Taylor expanded in t around 0 54.1%

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

      1. associate-/l* 54.5%

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

      2. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in k around 0 47.8%

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

      1. associate-/l* 49.1%

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

   10. Simplified 49.1%

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

   if 2.8000000000000001e-67 < t

   1. Initial program 64.2%

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

   3. Simplified 71.1%

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

      1. associate-/l/ 71.1%

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

      2. add-cube-cbrt 70.9%

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

      3. times-frac 70.9%

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

   6. Applied egg-rr 93.7%

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

      1. associate-/l/ 93.7%

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

   8. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{2}{\sin k}}}{\tan k \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}} \]

   9. Taylor expanded in k around 0 49.6%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   10. Step-by-step derivation

       1. *-commutative 49.6%

          \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

       2. associate-/r* 52.0%

          \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]

   11. Simplified 52.0%

       \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 52.0%

          \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]

   13. Applied egg-rr 52.0%

       \[\leadsto \frac{\frac{{\ell}^{2}}{{t}^{3}}}{\color{blue}{k \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. unpow3 52.5%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. times-frac 65.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. pow2 65.6%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

4. Applied egg-rr 65.6%

   \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

5. Taylor expanded in t around 0 51.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
6. Step-by-step derivation

   1. associate-/l* 52.7%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   2. *-commutative 52.7%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

7. Simplified 52.7%

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

8. Taylor expanded in k around 0 45.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
9. Step-by-step derivation

   1. associate-/l* 46.9%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

10. Simplified 46.9%

    \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

11. Taylor expanded in k around 0 45.7%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

12. Final simplification 45.7%

    \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
13. Add Preprocessing

Alternative 20: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (pow k 4.0) (/ t_m (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 / (pow(k, 4.0) * (t_m / 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 / ((k ** 4.0d0) * (t_m / (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 / (Math.pow(k, 4.0) * (t_m / 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 / (math.pow(k, 4.0) * (t_m / 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((k ^ 4.0) * Float64(t_m / (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 / ((k ^ 4.0) * (t_m / (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[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. unpow3 52.5%

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. times-frac 65.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. pow2 65.6%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{{t}^{2}}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

4. Applied egg-rr 65.6%

   \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

5. Taylor expanded in t around 0 51.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
6. Step-by-step derivation

   1. associate-/l* 52.7%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   2. *-commutative 52.7%

      \[\leadsto \frac{2}{{k}^{2} \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]

7. Simplified 52.7%

   \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2} \cdot \cos k}}} \]

8. Taylor expanded in k around 0 45.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
9. Step-by-step derivation

   1. associate-/l* 46.9%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

10. Simplified 46.9%

    \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

11. Final simplification 46.9%

    \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))))