Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.5% → 83.9%

Time: 33.1s

Alternatives: 27

Speedup: 1.9×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k}\\ t_3 := \tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\\ t_4 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t_5 := t\_4 \cdot \left(\frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot t\_4\right)\\ t_6 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.42 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{t\_6} \cdot t\_2\right)}^{3} \cdot t\_3}\\ \mathbf{elif}\;t\_m \leq 8.2 \cdot 10^{+30}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot {\left(\frac{t\_m \cdot t\_2}{t\_6}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (sin k)))
        (t_3 (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
        (t_4 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m)))))
        (t_5 (* t_4 (* (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))) t_4)))
        (t_6 (pow (cbrt l) 2.0)))
   (*
    t_s
    (if (<= t_m 1.42e-102)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (if (<= t_m 4.8e-48)
        t_5
        (if (<= t_m 5e-22)
          (/ 2.0 (* (pow (* (/ t_m t_6) t_2) 3.0) t_3))
          (if (<= t_m 8.2e+30)
            t_5
            (/ 2.0 (* t_3 (pow (/ (* t_m t_2) t_6) 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(sin(k));
	double t_3 = tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0));
	double t_4 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double t_5 = t_4 * ((2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * t_4);
	double t_6 = pow(cbrt(l), 2.0);
	double tmp;
	if (t_m <= 1.42e-102) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 4.8e-48) {
		tmp = t_5;
	} else if (t_m <= 5e-22) {
		tmp = 2.0 / (pow(((t_m / t_6) * t_2), 3.0) * t_3);
	} else if (t_m <= 8.2e+30) {
		tmp = t_5;
	} else {
		tmp = 2.0 / (t_3 * pow(((t_m * t_2) / t_6), 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(Math.sin(k));
	double t_3 = Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0));
	double t_4 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double t_5 = t_4 * ((2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * t_4);
	double t_6 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t_m <= 1.42e-102) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 4.8e-48) {
		tmp = t_5;
	} else if (t_m <= 5e-22) {
		tmp = 2.0 / (Math.pow(((t_m / t_6) * t_2), 3.0) * t_3);
	} else if (t_m <= 8.2e+30) {
		tmp = t_5;
	} else {
		tmp = 2.0 / (t_3 * Math.pow(((t_m * t_2) / t_6), 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(sin(k))
	t_3 = Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))
	t_4 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	t_5 = Float64(t_4 * Float64(Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * t_4))
	t_6 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.42e-102)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 4.8e-48)
		tmp = t_5;
	elseif (t_m <= 5e-22)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / t_6) * t_2) ^ 3.0) * t_3));
	elseif (t_m <= 8.2e+30)
		tmp = t_5;
	else
		tmp = Float64(2.0 / Float64(t_3 * (Float64(Float64(t_m * t_2) / t_6) ^ 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[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[(N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.42e-102], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e-48], t$95$5, If[LessEqual[t$95$m, 5e-22], N[(2.0 / N[(N[Power[N[(N[(t$95$m / t$95$6), $MachinePrecision] * t$95$2), $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8.2e+30], t$95$5, N[(2.0 / N[(t$95$3 * N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] / t$95$6), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.42000000000000009e-102

   1. Initial program 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in t around 0 61.9%

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

      1. associate-*r* 61.9%

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

   7. Simplified 61.9%

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

   if 1.42000000000000009e-102 < t < 4.8e-48 or 4.99999999999999954e-22 < t < 8.20000000000000011e30

   1. Initial program 61.1%

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

   3. Simplified 57.2%

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

      1. associate-*r* 61.0%

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

      2. add-sqr-sqrt 61.0%

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

      3. times-frac 64.8%

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

   6. Applied egg-rr 84.3%

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

      1. associate-/l* 88.3%

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

      2. associate-*l* 88.3%

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

   8. Simplified 88.3%

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

   if 4.8e-48 < t < 4.99999999999999954e-22

   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 67.5%

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

      1. add-cube-cbrt 67.5%

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

      2. pow3 67.5%

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

      3. *-commutative 67.5%

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

      4. cbrt-prod 67.5%

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

      5. cbrt-div 67.5%

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

      6. rem-cbrt-cube 67.5%

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

      7. cbrt-prod 98.3%

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

      8. pow2 98.3%

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

   6. Applied egg-rr 98.3%

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

   if 8.20000000000000011e30 < t

   1. Initial program 67.0%

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

   3. Simplified 67.0%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. *-commutative 67.0%

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

      4. cbrt-prod 67.0%

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

      5. cbrt-div 68.8%

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

      6. rem-cbrt-cube 73.3%

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

      7. cbrt-prod 95.5%

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

      8. pow2 95.5%

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

   6. Applied egg-rr 95.5%

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

      1. associate-*r/ 95.6%

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

   8. Applied egg-rr 95.6%

      \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.42 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\\ t_3 := t\_2 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-269}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t\_2\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
        (t_3 (* t_2 (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))))
   (*
    t_s
    (if (<= t_3 5e-269)
      (/ 2.0 (* (* (tan k) t_2) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (if (<= t_3 INFINITY)
        (/ 2.0 (pow (* (sqrt 2.0) (* k (/ (pow t_m 1.5) l))) 2.0))
        (*
         2.0
         (*
          (/ (pow l 2.0) (pow k 2.0))
          (/ (cos k) (* t_m (pow (sin k) 2.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (pow((k / t_m), 2.0) + 1.0);
	double t_3 = t_2 * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_3 <= 5e-269) {
		tmp = 2.0 / ((tan(k) * t_2) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = 2.0 / pow((sqrt(2.0) * (k * (pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (Math.pow((k / t_m), 2.0) + 1.0);
	double t_3 = t_2 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_3 <= 5e-269) {
		tmp = 2.0 / ((Math.tan(k) * t_2) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 1.0 + (math.pow((k / t_m), 2.0) + 1.0)
	t_3 = t_2 * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	tmp = 0
	if t_3 <= 5e-269:
		tmp = 2.0 / ((math.tan(k) * t_2) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	elif t_3 <= math.inf:
		tmp = 2.0 / math.pow((math.sqrt(2.0) * (k * (math.pow(t_m, 1.5) / l))), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))
	t_3 = Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))
	tmp = 0.0
	if (t_3 <= 5e-269)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * t_2) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	elseif (t_3 <= Inf)
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 1.0 + (((k / t_m) ^ 2.0) + 1.0);
	t_3 = t_2 * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	tmp = 0.0;
	if (t_3 <= 5e-269)
		tmp = 2.0 / ((tan(k) * t_2) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	elseif (t_3 <= Inf)
		tmp = 2.0 / ((sqrt(2.0) * (k * ((t_m ^ 1.5) / l))) ^ 2.0);
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (cos(k) / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 5e-269], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 4.99999999999999979e-269

   1. Initial program 82.3%

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

   3. Simplified 82.2%

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

      1. unpow3 82.2%

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

      2. times-frac 88.3%

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

      3. pow2 88.3%

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

   6. Applied egg-rr 88.3%

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

   if 4.99999999999999979e-269 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 87.0%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in k around 0 78.0%

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

      1. add-sqr-sqrt 49.0%

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

      2. pow2 49.0%

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

      3. associate-*r* 49.0%

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

      4. sqrt-prod 49.0%

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

      5. associate-/l/ 47.2%

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

      6. unpow2 47.2%

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

      7. div-inv 47.2%

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

      8. pow-flip 47.2%

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

      9. metadata-eval 47.2%

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

      10. sqrt-pow1 53.1%

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

      11. metadata-eval 53.1%

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

      12. pow1 53.1%

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

   7. Applied egg-rr 53.1%

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

      1. pow1 53.1%

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

      2. *-commutative 53.1%

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

      3. sqrt-prod 53.1%

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

      4. sqrt-prod 55.0%

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

      5. sqrt-pow1 56.9%

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

      6. metadata-eval 56.9%

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

      7. sqrt-pow1 60.0%

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

      8. metadata-eval 60.0%

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

      9. unpow-1 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. unpow1 60.0%

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

       2. *-commutative 60.0%

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

       3. *-commutative 60.0%

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

       4. associate-*l* 60.0%

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

       5. associate-*r/ 60.0%

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

       6. *-rgt-identity 60.0%

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

   11. Simplified 60.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in t around 0 47.0%

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

      1. times-frac 44.9%

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

   7. Simplified 44.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq 5 \cdot 10^{-269}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq \infty:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.08 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.08e-106)
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (* (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))) (cbrt (sin k))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.08e-106) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))) * cbrt(sin(k)))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.08e-106) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * Math.cbrt(Math.sin(k)))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.08e-106)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * cbrt(sin(k)))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.08e-106], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.08e-106

   1. Initial program 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in t around 0 61.9%

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

      1. associate-*r* 61.9%

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

   7. Simplified 61.9%

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

   if 1.08e-106 < t

   1. Initial program 65.2%

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

   3. Simplified 65.2%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

      2. metadata-eval 80.2%

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

      3. associate-+r+ 80.2%

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

      4. cbrt-prod 90.6%

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

      5. associate-+r+ 90.6%

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

      6. metadata-eval 90.6%

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

   8. Applied egg-rr 90.6%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := {\sin k}^{2}\\ t_4 := \sin k \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{t\_2} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.76 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{{\left(\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\ell}\right) \cdot \sqrt{t\_4}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot \frac{t\_3}{\cos k}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot t\_3}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot t\_4}}{t\_2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0))
        (t_3 (pow (sin k) 2.0))
        (t_4 (* (sin k) (tan k))))
   (*
    t_s
    (if (<= k 1.5e-146)
      (/ 2.0 (pow (* (/ t_m t_2) (* (cbrt k) (* (cbrt k) (cbrt 2.0)))) 3.0))
      (if (<= k 1.76e-10)
        (/
         2.0
         (pow
          (*
           (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (/ (pow t_m 1.5) l))
           (sqrt t_4))
          2.0))
        (if (<= k 2.05e-7)
          (/
           2.0
           (pow
            (* (* t_m (pow (cbrt l) -2.0)) (cbrt (* 2.0 (/ t_3 (cos k)))))
            3.0))
          (if (<= k 1.95e+150)
            (/ 2.0 (/ (* (* t_m (pow k 2.0)) t_3) (* (pow l 2.0) (cos k))))
            (/
             2.0
             (pow
              (* t_m (/ (cbrt (* (+ 2.0 (pow (/ k t_m) 2.0)) t_4)) 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 = pow(cbrt(l), 2.0);
	double t_3 = pow(sin(k), 2.0);
	double t_4 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.5e-146) {
		tmp = 2.0 / pow(((t_m / t_2) * (cbrt(k) * (cbrt(k) * cbrt(2.0)))), 3.0);
	} else if (k <= 1.76e-10) {
		tmp = 2.0 / pow(((hypot(1.0, hypot(1.0, (k / t_m))) * (pow(t_m, 1.5) / l)) * sqrt(t_4)), 2.0);
	} else if (k <= 2.05e-7) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((2.0 * (t_3 / cos(k))))), 3.0);
	} else if (k <= 1.95e+150) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * t_3) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / pow((t_m * (cbrt(((2.0 + pow((k / t_m), 2.0)) * t_4)) / 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.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.pow(Math.sin(k), 2.0);
	double t_4 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.5e-146) {
		tmp = 2.0 / Math.pow(((t_m / t_2) * (Math.cbrt(k) * (Math.cbrt(k) * Math.cbrt(2.0)))), 3.0);
	} else if (k <= 1.76e-10) {
		tmp = 2.0 / Math.pow(((Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * (Math.pow(t_m, 1.5) / l)) * Math.sqrt(t_4)), 2.0);
	} else if (k <= 2.05e-7) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((2.0 * (t_3 / Math.cos(k))))), 3.0);
	} else if (k <= 1.95e+150) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * t_3) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.cbrt(((2.0 + Math.pow((k / t_m), 2.0)) * t_4)) / 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(l) ^ 2.0
	t_3 = sin(k) ^ 2.0
	t_4 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.5e-146)
		tmp = Float64(2.0 / (Float64(Float64(t_m / t_2) * Float64(cbrt(k) * Float64(cbrt(k) * cbrt(2.0)))) ^ 3.0));
	elseif (k <= 1.76e-10)
		tmp = Float64(2.0 / (Float64(Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * Float64((t_m ^ 1.5) / l)) * sqrt(t_4)) ^ 2.0));
	elseif (k <= 2.05e-7)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(2.0 * Float64(t_3 / cos(k))))) ^ 3.0));
	elseif (k <= 1.95e+150)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * t_3) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(cbrt(Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * t_4)) / 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[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.5e-146], N[(2.0 / N[Power[N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.76e-10], N[(2.0 / N[Power[N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.05e-7], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[(t$95$3 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.95e+150], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], 1/3], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if k < 1.50000000000000009e-146

   1. Initial program 50.6%

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

   3. Simplified 50.6%

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

      1. add-cube-cbrt 50.5%

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

      2. pow3 50.5%

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

   6. Applied egg-rr 65.4%

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

      1. *-commutative 65.4%

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

      2. metadata-eval 65.4%

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

      3. associate-+r+ 65.4%

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

      4. cbrt-prod 80.1%

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

      5. associate-+r+ 80.1%

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

      6. metadata-eval 80.1%

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

   8. Applied egg-rr 80.1%

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

   9. Taylor expanded in k around 0 73.4%

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

   10. Taylor expanded in k around 0 71.5%

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

   if 1.50000000000000009e-146 < k < 1.7600000000000001e-10

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   6. Applied egg-rr 53.9%

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

      1. associate-*r* 53.9%

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

   8. Simplified 53.9%

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

   if 1.7600000000000001e-10 < k < 2.05e-7

   1. Initial program 0.0%

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

   3. Simplified 2.2%

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

      1. add-cube-cbrt 2.2%

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

      2. pow3 2.2%

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

      3. cbrt-div 2.2%

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

      4. rem-cbrt-cube 2.2%

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

   6. Applied egg-rr 2.2%

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

      1. add-cube-cbrt 2.2%

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

      2. pow3 2.2%

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

   8. Applied egg-rr 49.2%

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

   9. Taylor expanded in t around inf 99.2%

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

   if 2.05e-7 < k < 1.94999999999999995e150

   1. Initial program 37.8%

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

   3. Simplified 37.8%

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

   5. Taylor expanded in t around 0 71.1%

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

      1. associate-*r* 71.2%

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

   7. Simplified 71.2%

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

   if 1.94999999999999995e150 < k

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

      1. add-cube-cbrt 37.1%

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

      2. pow3 37.1%

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

   6. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.9%

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

   8. Applied egg-rr 59.9%

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

      1. associate-/l* 59.9%

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

      2. associate-*r* 59.9%

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

   10. Simplified 59.9%

       \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \frac{\sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.76 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{{\left(\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot \frac{{\sin k}^{2}}{\cos k}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t_4 := \sin k \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{t\_2} \cdot \left(\sqrt[3]{\tan k \cdot t\_3} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\ell}\right) \cdot \sqrt{t\_4}\right)}^{2}}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \frac{\sqrt[3]{t\_3 \cdot t\_4}}{t\_2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0))
        (t_3 (+ 2.0 (pow (/ k t_m) 2.0)))
        (t_4 (* (sin k) (tan k))))
   (*
    t_s
    (if (<= k 7.2e-94)
      (/ 2.0 (pow (* (/ t_m t_2) (* (cbrt (* (tan k) t_3)) (cbrt k))) 3.0))
      (if (<= k 2.5e-16)
        (/
         2.0
         (pow
          (*
           (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (/ (pow t_m 1.5) l))
           (sqrt t_4))
          2.0))
        (if (<= k 7.6e+99)
          (/
           2.0
           (/
            (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))
            (* (pow l 2.0) (cos k))))
          (/ 2.0 (pow (* t_m (/ (cbrt (* t_3 t_4)) 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 = pow(cbrt(l), 2.0);
	double t_3 = 2.0 + pow((k / t_m), 2.0);
	double t_4 = sin(k) * tan(k);
	double tmp;
	if (k <= 7.2e-94) {
		tmp = 2.0 / pow(((t_m / t_2) * (cbrt((tan(k) * t_3)) * cbrt(k))), 3.0);
	} else if (k <= 2.5e-16) {
		tmp = 2.0 / pow(((hypot(1.0, hypot(1.0, (k / t_m))) * (pow(t_m, 1.5) / l)) * sqrt(t_4)), 2.0);
	} else if (k <= 7.6e+99) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / pow((t_m * (cbrt((t_3 * t_4)) / 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.pow(Math.cbrt(l), 2.0);
	double t_3 = 2.0 + Math.pow((k / t_m), 2.0);
	double t_4 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 7.2e-94) {
		tmp = 2.0 / Math.pow(((t_m / t_2) * (Math.cbrt((Math.tan(k) * t_3)) * Math.cbrt(k))), 3.0);
	} else if (k <= 2.5e-16) {
		tmp = 2.0 / Math.pow(((Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * (Math.pow(t_m, 1.5) / l)) * Math.sqrt(t_4)), 2.0);
	} else if (k <= 7.6e+99) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.cbrt((t_3 * t_4)) / 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(l) ^ 2.0
	t_3 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	t_4 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 7.2e-94)
		tmp = Float64(2.0 / (Float64(Float64(t_m / t_2) * Float64(cbrt(Float64(tan(k) * t_3)) * cbrt(k))) ^ 3.0));
	elseif (k <= 2.5e-16)
		tmp = Float64(2.0 / (Float64(Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * Float64((t_m ^ 1.5) / l)) * sqrt(t_4)) ^ 2.0));
	elseif (k <= 7.6e+99)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(cbrt(Float64(t_3 * t_4)) / 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[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 7.2e-94], N[(2.0 / N[Power[N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[(N[Power[N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e-16], N[(2.0 / N[Power[N[(N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.6e+99], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[(t$95$3 * t$95$4), $MachinePrecision], 1/3], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 7.2e-94

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

      1. add-cube-cbrt 53.4%

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

      2. pow3 53.4%

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

   6. Applied egg-rr 68.3%

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

      1. *-commutative 68.3%

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

      2. metadata-eval 68.3%

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

      3. associate-+r+ 68.3%

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

      4. cbrt-prod 81.7%

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

      5. associate-+r+ 81.7%

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

      6. metadata-eval 81.7%

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

   8. Applied egg-rr 81.7%

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

   9. Taylor expanded in k around 0 75.6%

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

   if 7.2e-94 < k < 2.5000000000000002e-16

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

   6. Applied egg-rr 52.0%

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

      1. associate-*r* 52.0%

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

   8. Simplified 52.0%

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

   if 2.5000000000000002e-16 < k < 7.6e99

   1. Initial program 47.3%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. associate-*r* 73.3%

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

   7. Simplified 73.3%

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

   if 7.6e99 < k

   1. Initial program 30.8%

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

   3. Simplified 30.8%

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

      1. add-cube-cbrt 30.8%

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

      2. pow3 30.8%

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

   6. Applied egg-rr 58.7%

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

      1. associate-*l/ 58.8%

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

   8. Applied egg-rr 58.8%

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

      1. associate-/l* 58.8%

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

      2. associate-*r* 58.8%

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

   10. Simplified 58.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right) \cdot {\left(t\_m \cdot \left(\sqrt[3]{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t\_m \cdot t\_3}\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+176}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{t\_2} \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \frac{\sqrt[3]{\frac{2 \cdot t\_3}{\cos k}}}{t\_2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0)) (t_3 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= l 5.8e+51)
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
        (pow (* t_m (* (cbrt k) (pow (cbrt l) -2.0))) 3.0)))
      (if (<= l 3.3e+72)
        (* 2.0 (* (/ (pow l 2.0) (pow k 2.0)) (/ (cos k) (* t_m t_3))))
        (if (<= l 3.5e+176)
          (/
           2.0
           (pow
            (* (/ t_m t_2) (* (cbrt (sin k)) (* (cbrt k) (cbrt 2.0))))
            3.0))
          (/
           2.0
           (pow (* t_m (/ (cbrt (/ (* 2.0 t_3) (cos k))) 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 = pow(cbrt(l), 2.0);
	double t_3 = pow(sin(k), 2.0);
	double tmp;
	if (l <= 5.8e+51) {
		tmp = 2.0 / ((tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))) * pow((t_m * (cbrt(k) * pow(cbrt(l), -2.0))), 3.0));
	} else if (l <= 3.3e+72) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (cos(k) / (t_m * t_3)));
	} else if (l <= 3.5e+176) {
		tmp = 2.0 / pow(((t_m / t_2) * (cbrt(sin(k)) * (cbrt(k) * cbrt(2.0)))), 3.0);
	} else {
		tmp = 2.0 / pow((t_m * (cbrt(((2.0 * t_3) / cos(k))) / 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.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (l <= 5.8e+51) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))) * Math.pow((t_m * (Math.cbrt(k) * Math.pow(Math.cbrt(l), -2.0))), 3.0));
	} else if (l <= 3.3e+72) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (Math.cos(k) / (t_m * t_3)));
	} else if (l <= 3.5e+176) {
		tmp = 2.0 / Math.pow(((t_m / t_2) * (Math.cbrt(Math.sin(k)) * (Math.cbrt(k) * Math.cbrt(2.0)))), 3.0);
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.cbrt(((2.0 * t_3) / Math.cos(k))) / 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(l) ^ 2.0
	t_3 = sin(k) ^ 2.0
	tmp = 0.0
	if (l <= 5.8e+51)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))) * (Float64(t_m * Float64(cbrt(k) * (cbrt(l) ^ -2.0))) ^ 3.0)));
	elseif (l <= 3.3e+72)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(cos(k) / Float64(t_m * t_3))));
	elseif (l <= 3.5e+176)
		tmp = Float64(2.0 / (Float64(Float64(t_m / t_2) * Float64(cbrt(sin(k)) * Float64(cbrt(k) * cbrt(2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(cbrt(Float64(Float64(2.0 * t_3) / cos(k))) / 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[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 5.8e+51], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.3e+72], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.5e+176], N[(2.0 / N[Power[N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[(N[(2.0 * t$95$3), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if l < 5.7999999999999997e51

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in k around 0 51.7%

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

      1. associate-/l* 52.0%

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

   7. Simplified 52.0%

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

      1. add-cube-cbrt 52.0%

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

      2. pow3 52.0%

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

      3. unpow2 52.0%

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

      4. associate-/l/ 56.4%

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

      5. *-commutative 56.4%

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

      6. cbrt-prod 56.3%

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

      7. associate-/l/ 52.0%

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

      8. unpow2 52.0%

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

      9. cbrt-div 52.5%

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

      10. rem-cbrt-cube 63.2%

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

      11. unpow2 63.2%

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

      12. cbrt-prod 71.9%

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

      13. unpow2 71.9%

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

      14. div-inv 71.9%

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

      15. pow-flip 71.9%

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

      16. metadata-eval 71.9%

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

   9. Applied egg-rr 71.9%

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

       1. associate-*l* 72.0%

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

   11. Simplified 72.0%

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

   if 5.7999999999999997e51 < l < 3.3e72

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in t around 0 67.8%

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

      1. times-frac 67.7%

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

   7. Simplified 67.7%

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

   if 3.3e72 < l < 3.50000000000000003e176

   1. Initial program 37.5%

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

   3. Simplified 37.5%

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

      1. add-cube-cbrt 37.4%

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

      2. pow3 37.4%

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

   6. Applied egg-rr 48.9%

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

      1. *-commutative 48.9%

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

      2. metadata-eval 48.9%

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

      3. associate-+r+ 48.9%

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

      4. cbrt-prod 67.9%

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

      5. associate-+r+ 67.9%

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

      6. metadata-eval 67.9%

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

   8. Applied egg-rr 67.9%

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

   9. Taylor expanded in k around 0 73.5%

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

   if 3.50000000000000003e176 < l

   1. Initial program 20.8%

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

   3. Simplified 20.8%

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

      1. add-cube-cbrt 20.8%

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

      2. pow3 20.8%

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

   6. Applied egg-rr 58.0%

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

      1. associate-*l/ 58.4%

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

   8. Applied egg-rr 58.4%

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

      1. associate-/l* 58.5%

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

      2. associate-*r* 58.5%

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

   10. Simplified 58.5%

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

   11. Taylor expanded in t around inf 70.1%

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

       1. *-commutative 70.1%

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

       2. associate-*l/ 70.1%

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

   13. Simplified 70.1%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot {\left(t \cdot \left(\sqrt[3]{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+176}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{\frac{2 \cdot {\sin k}^{2}}{\cos k}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-107)
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
    (/
     2.0
     (*
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 3.0)
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-107) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-107) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.5e-107)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-107], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.49999999999999956e-107

   1. Initial program 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in t around 0 61.7%

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

      1. associate-*r* 61.7%

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

   7. Simplified 61.7%

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

   if 8.49999999999999956e-107 < t

   1. Initial program 64.4%

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

   3. Simplified 64.4%

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

      1. add-cube-cbrt 64.4%

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

      2. pow3 64.4%

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

      3. *-commutative 64.4%

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

      4. cbrt-prod 64.3%

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

      5. cbrt-div 65.4%

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

      6. rem-cbrt-cube 68.3%

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

      7. cbrt-prod 86.1%

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

      8. pow2 86.1%

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

   6. Applied egg-rr 86.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\\ t_3 := \frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ t_4 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(t\_4 \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot t\_2}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+101}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{{t\_4}^{3} \cdot t\_2}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t\_m \cdot k}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
        (t_3
         (/
          2.0
          (/
           (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))
           (* (pow l 2.0) (cos k)))))
        (t_4 (/ t_m (pow (cbrt l) 2.0))))
   (*
    t_s
    (if (<= k 6.2e-8)
      (/ 2.0 (pow (* t_4 (* (cbrt k) (* (cbrt k) (cbrt 2.0)))) 3.0))
      (if (<= k 2.6e+15)
        (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) t_2))
        (if (<= k 9e+101)
          t_3
          (if (<= k 6.5e+110)
            (/ 2.0 (* (pow t_4 3.0) t_2))
            (if (<= k 2e+152)
              t_3
              (/
               2.0
               (pow
                (* (* k (cbrt (pow l -2.0))) (cbrt (* t_m k)))
                3.0))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)));
	double t_3 = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	double t_4 = t_m / pow(cbrt(l), 2.0);
	double tmp;
	if (k <= 6.2e-8) {
		tmp = 2.0 / pow((t_4 * (cbrt(k) * (cbrt(k) * cbrt(2.0)))), 3.0);
	} else if (k <= 2.6e+15) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * t_2);
	} else if (k <= 9e+101) {
		tmp = t_3;
	} else if (k <= 6.5e+110) {
		tmp = 2.0 / (pow(t_4, 3.0) * t_2);
	} else if (k <= 2e+152) {
		tmp = t_3;
	} else {
		tmp = 2.0 / pow(((k * cbrt(pow(l, -2.0))) * cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)));
	double t_3 = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	double t_4 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k <= 6.2e-8) {
		tmp = 2.0 / Math.pow((t_4 * (Math.cbrt(k) * (Math.cbrt(k) * Math.cbrt(2.0)))), 3.0);
	} else if (k <= 2.6e+15) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * t_2);
	} else if (k <= 9e+101) {
		tmp = t_3;
	} else if (k <= 6.5e+110) {
		tmp = 2.0 / (Math.pow(t_4, 3.0) * t_2);
	} else if (k <= 2e+152) {
		tmp = t_3;
	} else {
		tmp = 2.0 / Math.pow(((k * Math.cbrt(Math.pow(l, -2.0))) * Math.cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))
	t_3 = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))))
	t_4 = Float64(t_m / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (k <= 6.2e-8)
		tmp = Float64(2.0 / (Float64(t_4 * Float64(cbrt(k) * Float64(cbrt(k) * cbrt(2.0)))) ^ 3.0));
	elseif (k <= 2.6e+15)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * t_2));
	elseif (k <= 9e+101)
		tmp = t_3;
	elseif (k <= 6.5e+110)
		tmp = Float64(2.0 / Float64((t_4 ^ 3.0) * t_2));
	elseif (k <= 2e+152)
		tmp = t_3;
	else
		tmp = Float64(2.0 / (Float64(Float64(k * cbrt((l ^ -2.0))) * cbrt(Float64(t_m * k))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6.2e-8], N[(2.0 / N[Power[N[(t$95$4 * N[(N[Power[k, 1/3], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.6e+15], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9e+101], t$95$3, If[LessEqual[k, 6.5e+110], N[(2.0 / N[(N[Power[t$95$4, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e+152], t$95$3, N[(2.0 / N[Power[N[(N[(k * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if k < 6.2e-8

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

      1. add-cube-cbrt 54.0%

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

      2. pow3 54.0%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. metadata-eval 69.2%

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

      3. associate-+r+ 69.2%

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

      4. cbrt-prod 80.7%

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

      5. associate-+r+ 80.7%

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

      6. metadata-eval 80.7%

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

   8. Applied egg-rr 80.7%

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

   9. Taylor expanded in k around 0 75.4%

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

   10. Taylor expanded in k around 0 74.7%

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

   if 6.2e-8 < k < 2.6e15

   1. Initial program 45.3%

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

   3. Simplified 56.4%

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

      1. add-cube-cbrt 56.3%

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

      2. pow3 56.1%

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

      3. cbrt-div 56.1%

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

      4. rem-cbrt-cube 56.1%

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

   6. Applied egg-rr 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 56.0%

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

   8. Applied egg-rr 88.6%

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

      1. cube-prod 88.5%

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

      2. rem-cube-cbrt 88.5%

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

      3. *-commutative 88.5%

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

   10. Simplified 88.5%

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

   if 2.6e15 < k < 9.0000000000000004e101 or 6.4999999999999997e110 < k < 2.0000000000000001e152

   1. Initial program 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in t around 0 79.2%

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

      1. associate-*r* 79.2%

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

   7. Simplified 79.2%

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

   if 9.0000000000000004e101 < k < 6.4999999999999997e110

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. add-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

   6. Applied egg-rr 98.8%

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

      1. cube-prod 98.8%

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

      2. rem-cube-cbrt 98.9%

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

   8. Simplified 98.9%

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

   if 2.0000000000000001e152 < k

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0 37.4%

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

      1. associate-/l* 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in k around inf 56.7%

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

      1. times-frac 56.7%

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

   10. Simplified 56.7%

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

       1. add-cube-cbrt 56.7%

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

       2. pow3 56.7%

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

       3. cbrt-prod 56.7%

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

       4. div-inv 56.7%

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

       5. cbrt-prod 56.7%

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

       6. unpow3 56.7%

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

       7. add-cbrt-cube 59.6%

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

       8. pow-flip 59.6%

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

       9. metadata-eval 59.6%

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

       10. associate-/l* 59.6%

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

       11. tan-quot 59.6%

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

   12. Applied egg-rr 59.6%

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

   13. Taylor expanded in k around 0 59.7%

       \[\leadsto \frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \color{blue}{\sqrt[3]{k \cdot t}}\right)}^{3}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t \cdot k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + t\_2\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2 \cdot \sin k}\right)\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 4.6e-89)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (if (<= t_m 4.2e-39)
        (/
         2.0
         (*
          (/ 1.0 l)
          (* (* (tan k) (+ 2.0 t_2)) (* (sin k) (/ (pow t_m 3.0) l)))))
        (if (<= t_m 1.06e+100)
          (/
           2.0
           (*
            (* (tan k) (+ 1.0 (+ t_2 1.0)))
            (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
          (/
           2.0
           (pow
            (* t_m (* (pow (cbrt l) -2.0) (* (cbrt k) (cbrt (* 2.0 (sin k))))))
            3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.6e-89) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 4.2e-39) {
		tmp = 2.0 / ((1.0 / l) * ((tan(k) * (2.0 + t_2)) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.06e+100) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt(k) * cbrt((2.0 * sin(k)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.6e-89) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 4.2e-39) {
		tmp = 2.0 / ((1.0 / l) * ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else if (t_m <= 1.06e+100) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt(k) * Math.cbrt((2.0 * Math.sin(k)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 4.6e-89)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 4.2e-39)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	elseif (t_m <= 1.06e+100)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(k) * cbrt(Float64(2.0 * sin(k)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.6e-89], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-39], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.06e+100], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.6e-89

   1. Initial program 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 62.3%

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

      1. associate-*r* 62.4%

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

   7. Simplified 62.4%

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

   if 4.6e-89 < t < 4.19999999999999987e-39

   1. Initial program 54.8%

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

   3. Simplified 54.7%

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

      1. associate-*l* 54.6%

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

      2. associate-/r* 67.3%

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

      3. associate-+r+ 67.3%

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

      4. metadata-eval 67.3%

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

      5. associate-*l* 67.4%

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

      6. associate-*l/ 68.2%

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

      7. clear-num 68.2%

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

      8. associate-*l* 68.2%

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

   6. Applied egg-rr 68.2%

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

      1. associate-/r/ 68.2%

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

      2. associate-*r* 68.1%

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

   8. Simplified 68.1%

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

   if 4.19999999999999987e-39 < t < 1.06000000000000007e100

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

      1. unpow3 80.1%

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

      2. times-frac 82.9%

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

      3. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   if 1.06000000000000007e100 < t

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

      1. add-cube-cbrt 61.3%

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

      2. pow3 61.3%

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

   6. Applied egg-rr 82.5%

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

      1. *-commutative 82.5%

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

      2. metadata-eval 82.5%

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

      3. associate-+r+ 82.5%

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

      4. cbrt-prod 98.8%

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

      5. associate-+r+ 98.8%

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

      6. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

   9. Taylor expanded in k around 0 83.7%

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

       1. pow1 83.7%

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

       2. div-inv 83.7%

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

       3. pow-flip 83.7%

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

       4. metadata-eval 83.7%

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

       5. cbrt-unprod 83.6%

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

       6. cbrt-unprod 67.4%

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

   11. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

       2. associate-*l* 67.5%

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

       3. *-commutative 67.5%

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

       4. *-commutative 67.5%

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

   13. Simplified 67.5%

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

       1. associate-*r* 67.5%

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

       2. cbrt-prod 82.0%

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

   15. Applied egg-rr 82.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2 \cdot \sin k}\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 5.9 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;\ell \cdot \frac{2}{{t\_m}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t\_m \cdot k}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.2e-8)
    (/
     2.0
     (pow
      (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt k) (* (cbrt k) (cbrt 2.0))))
      3.0))
    (if (<= k 7e+28)
      (/
       2.0
       (*
        (pow (* t_m (pow (cbrt l) -2.0)) 3.0)
        (* (sin k) (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      (if (<= k 5.9e+150)
        (/
         2.0
         (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
        (if (<= k 9.6e+150)
          (* l (/ 2.0 (* (pow t_m 3.0) (/ (* 2.0 (pow k 2.0)) l))))
          (/
           2.0
           (pow (* (* k (cbrt (pow l -2.0))) (cbrt (* t_m k))) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.2e-8) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(k) * (cbrt(k) * cbrt(2.0)))), 3.0);
	} else if (k <= 7e+28) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (sin(k) * (tan(k) * (2.0 + pow((k / t_m), 2.0)))));
	} else if (k <= 5.9e+150) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (k <= 9.6e+150) {
		tmp = l * (2.0 / (pow(t_m, 3.0) * ((2.0 * pow(k, 2.0)) / l)));
	} else {
		tmp = 2.0 / pow(((k * cbrt(pow(l, -2.0))) * cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.2e-8) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(k) * (Math.cbrt(k) * Math.cbrt(2.0)))), 3.0);
	} else if (k <= 7e+28) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (Math.sin(k) * (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))));
	} else if (k <= 5.9e+150) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (k <= 9.6e+150) {
		tmp = l * (2.0 / (Math.pow(t_m, 3.0) * ((2.0 * Math.pow(k, 2.0)) / l)));
	} else {
		tmp = 2.0 / Math.pow(((k * Math.cbrt(Math.pow(l, -2.0))) * Math.cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.2e-8)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(k) * Float64(cbrt(k) * cbrt(2.0)))) ^ 3.0));
	elseif (k <= 7e+28)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(sin(k) * Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))));
	elseif (k <= 5.9e+150)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (k <= 9.6e+150)
		tmp = Float64(l * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(Float64(2.0 * (k ^ 2.0)) / l))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * cbrt((l ^ -2.0))) * cbrt(Float64(t_m * k))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4.2e-8], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7e+28], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.9e+150], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.6e+150], N[(l * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if k < 4.19999999999999989e-8

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

      1. add-cube-cbrt 54.0%

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

      2. pow3 54.0%

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

   6. Applied egg-rr 69.2%

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

      1. *-commutative 69.2%

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

      2. metadata-eval 69.2%

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

      3. associate-+r+ 69.2%

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

      4. cbrt-prod 80.7%

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

      5. associate-+r+ 80.7%

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

      6. metadata-eval 80.7%

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

   8. Applied egg-rr 80.7%

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

   9. Taylor expanded in k around 0 75.4%

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

   10. Taylor expanded in k around 0 74.7%

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

   if 4.19999999999999989e-8 < k < 6.9999999999999999e28

   1. Initial program 43.8%

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

   3. Simplified 57.7%

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

      1. add-cube-cbrt 57.6%

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

      2. pow3 57.6%

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

      3. cbrt-div 57.3%

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

      4. rem-cbrt-cube 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. add-cube-cbrt 64.5%

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

      2. pow3 64.5%

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

   8. Applied egg-rr 85.3%

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

      1. cube-prod 85.1%

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

      2. rem-cube-cbrt 85.1%

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

      3. *-commutative 85.1%

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

   10. Simplified 85.1%

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

   if 6.9999999999999999e28 < k < 5.90000000000000023e150

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in t around 0 78.2%

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

      1. associate-*r* 78.2%

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

   7. Simplified 78.2%

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

   if 5.90000000000000023e150 < k < 9.60000000000000011e150

   1. Initial program 48.8%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in k around 0 51.3%

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

      1. associate-*l/ 52.1%

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

   7. Applied egg-rr 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. associate-/r/ 52.1%

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

   9. Applied egg-rr 52.1%

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

       1. *-lft-identity 52.1%

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

       2. *-commutative 52.1%

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

       3. associate-*l/ 51.0%

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

       4. associate-/l* 51.6%

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

   11. Simplified 51.6%

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

   if 9.60000000000000011e150 < k

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0 37.4%

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

      1. associate-/l* 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in k around inf 56.7%

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

      1. times-frac 56.7%

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

   10. Simplified 56.7%

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

       1. add-cube-cbrt 56.7%

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

       2. pow3 56.7%

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

       3. cbrt-prod 56.7%

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

       4. div-inv 56.7%

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

       5. cbrt-prod 56.7%

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

       6. unpow3 56.7%

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

       7. add-cbrt-cube 59.6%

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

       8. pow-flip 59.6%

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

       9. metadata-eval 59.6%

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

       10. associate-/l* 59.6%

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

       11. tan-quot 59.6%

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

   12. Applied egg-rr 59.6%

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

   13. Taylor expanded in k around 0 59.7%

       \[\leadsto \frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \color{blue}{\sqrt[3]{k \cdot t}}\right)}^{3}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 5.9 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{+150}:\\ \;\;\;\;\ell \cdot \frac{2}{{t}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t \cdot k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.3 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + t\_2\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \frac{\sqrt[3]{2} \cdot {\left(\sqrt[3]{k}\right)}^{2}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.3e-89)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (if (<= t_m 2.1e-39)
        (/
         2.0
         (*
          (/ 1.0 l)
          (* (* (tan k) (+ 2.0 t_2)) (* (sin k) (/ (pow t_m 3.0) l)))))
        (if (<= t_m 5.1e+100)
          (/
           2.0
           (*
            (* (tan k) (+ 1.0 (+ t_2 1.0)))
            (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
          (/
           2.0
           (pow
            (* t_m (/ (* (cbrt 2.0) (pow (cbrt k) 2.0)) (pow (cbrt l) 2.0)))
            3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.3e-89) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 2.1e-39) {
		tmp = 2.0 / ((1.0 / l) * ((tan(k) * (2.0 + t_2)) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else if (t_m <= 5.1e+100) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / pow((t_m * ((cbrt(2.0) * pow(cbrt(k), 2.0)) / pow(cbrt(l), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.3e-89) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 2.1e-39) {
		tmp = 2.0 / ((1.0 / l) * ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else if (t_m <= 5.1e+100) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / Math.pow((t_m * ((Math.cbrt(2.0) * Math.pow(Math.cbrt(k), 2.0)) / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.3e-89)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 2.1e-39)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	elseif (t_m <= 5.1e+100)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64(Float64(cbrt(2.0) * (cbrt(k) ^ 2.0)) / (cbrt(l) ^ 2.0))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.3e-89], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e-39], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.1e+100], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.2999999999999999e-89

   1. Initial program 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 62.3%

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

      1. associate-*r* 62.4%

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

   7. Simplified 62.4%

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

   if 1.2999999999999999e-89 < t < 2.09999999999999993e-39

   1. Initial program 54.8%

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

   3. Simplified 54.7%

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

      1. associate-*l* 54.6%

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

      2. associate-/r* 67.3%

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

      3. associate-+r+ 67.3%

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

      4. metadata-eval 67.3%

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

      5. associate-*l* 67.4%

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

      6. associate-*l/ 68.2%

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

      7. clear-num 68.2%

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

      8. associate-*l* 68.2%

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

   6. Applied egg-rr 68.2%

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

      1. associate-/r/ 68.2%

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

      2. associate-*r* 68.1%

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

   8. Simplified 68.1%

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

   if 2.09999999999999993e-39 < t < 5.1000000000000001e100

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

      1. unpow3 80.1%

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

      2. times-frac 82.9%

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

      3. pow2 82.9%

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

   6. Applied egg-rr 82.9%

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

   if 5.1000000000000001e100 < t

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

      1. add-cube-cbrt 61.3%

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

      2. pow3 61.3%

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

   6. Applied egg-rr 82.5%

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

      1. associate-*l/ 82.7%

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

   8. Applied egg-rr 82.7%

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

      1. associate-/l* 82.7%

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

      2. associate-*r* 82.7%

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

   10. Simplified 82.7%

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

   11. Taylor expanded in k around 0 67.3%

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

       1. *-commutative 67.3%

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

       2. cbrt-prod 67.3%

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

       3. unpow2 67.3%

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

       4. cbrt-prod 81.8%

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

       5. pow2 81.8%

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

   13. Applied egg-rr 81.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{2} \cdot {\left(\sqrt[3]{k}\right)}^{2}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5000000:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t\_m \cdot k}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5000000.0)
    (/
     2.0
     (pow
      (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt k) (* (cbrt k) (cbrt 2.0))))
      3.0))
    (if (<= k 5.8e+149)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (/ 2.0 (pow (* (* k (cbrt (pow l -2.0))) (cbrt (* t_m k))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5000000.0) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(k) * (cbrt(k) * cbrt(2.0)))), 3.0);
	} else if (k <= 5.8e+149) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / pow(((k * cbrt(pow(l, -2.0))) * cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5000000.0) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(k) * (Math.cbrt(k) * Math.cbrt(2.0)))), 3.0);
	} else if (k <= 5.8e+149) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow(((k * Math.cbrt(Math.pow(l, -2.0))) * Math.cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5000000.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(k) * Float64(cbrt(k) * cbrt(2.0)))) ^ 3.0));
	elseif (k <= 5.8e+149)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * cbrt((l ^ -2.0))) * cbrt(Float64(t_m * k))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5000000.0], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.8e+149], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5e6

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

      1. add-cube-cbrt 54.0%

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

      2. pow3 54.0%

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

   6. Applied egg-rr 69.8%

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

      1. *-commutative 69.8%

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

      2. metadata-eval 69.8%

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

      3. associate-+r+ 69.8%

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

      4. cbrt-prod 81.1%

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

      5. associate-+r+ 81.1%

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

      6. metadata-eval 81.1%

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

   8. Applied egg-rr 81.1%

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

   9. Taylor expanded in k around 0 75.7%

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

   10. Taylor expanded in k around 0 74.7%

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

   if 5e6 < k < 5.8000000000000004e149

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in t around 0 73.1%

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

      1. associate-*r* 73.1%

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

   7. Simplified 73.1%

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

   if 5.8000000000000004e149 < k

   1. Initial program 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in k around 0 36.4%

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

      1. associate-/l* 36.2%

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

   7. Simplified 36.2%

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

   8. Taylor expanded in k around inf 55.6%

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

      1. times-frac 55.6%

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

   10. Simplified 55.6%

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

       1. add-cube-cbrt 55.6%

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

       2. pow3 55.6%

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

       3. cbrt-prod 55.6%

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

       4. div-inv 55.6%

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

       5. cbrt-prod 55.6%

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

       6. unpow3 55.6%

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

       7. add-cbrt-cube 58.4%

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

       8. pow-flip 58.4%

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

       9. metadata-eval 58.4%

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

       10. associate-/l* 58.4%

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

       11. tan-quot 58.4%

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

   12. Applied egg-rr 58.4%

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

   13. Taylor expanded in k around 0 58.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5000000:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t \cdot k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + t\_2\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+167}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {t\_3}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot t\_3\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)) (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 2.4e-89)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (if (<= t_m 4.2e-39)
        (/
         2.0
         (*
          (/ 1.0 l)
          (* (* (tan k) (+ 2.0 t_2)) (* (sin k) (/ (pow t_m 3.0) l)))))
        (if (<= t_m 2.1e+167)
          (/ 2.0 (* (* (tan k) (+ 1.0 (+ t_2 1.0))) (* (sin k) (pow t_3 2.0))))
          (/ 2.0 (pow (* (sqrt 2.0) (* k t_3)) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 2.4e-89) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 4.2e-39) {
		tmp = 2.0 / ((1.0 / l) * ((tan(k) * (2.0 + t_2)) * (sin(k) * (pow(t_m, 3.0) / l))));
	} else if (t_m <= 2.1e+167) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * pow(t_3, 2.0)));
	} else {
		tmp = 2.0 / pow((sqrt(2.0) * (k * t_3)), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    t_3 = (t_m ** 1.5d0) / l
    if (t_m <= 2.4d-89) then
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) * (sin(k) ** 2.0d0)) / ((l ** 2.0d0) * cos(k)))
    else if (t_m <= 4.2d-39) then
        tmp = 2.0d0 / ((1.0d0 / l) * ((tan(k) * (2.0d0 + t_2)) * (sin(k) * ((t_m ** 3.0d0) / l))))
    else if (t_m <= 2.1d+167) then
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (t_2 + 1.0d0))) * (sin(k) * (t_3 ** 2.0d0)))
    else
        tmp = 2.0d0 / ((sqrt(2.0d0) * (k * t_3)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 2.4e-89) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 4.2e-39) {
		tmp = 2.0 / ((1.0 / l) * ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else if (t_m <= 2.1e+167) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * Math.pow(t_3, 2.0)));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * t_3)), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	t_3 = math.pow(t_m, 1.5) / l
	tmp = 0
	if t_m <= 2.4e-89:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) * math.pow(math.sin(k), 2.0)) / (math.pow(l, 2.0) * math.cos(k)))
	elif t_m <= 4.2e-39:
		tmp = 2.0 / ((1.0 / l) * ((math.tan(k) * (2.0 + t_2)) * (math.sin(k) * (math.pow(t_m, 3.0) / l))))
	elif t_m <= 2.1e+167:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (t_2 + 1.0))) * (math.sin(k) * math.pow(t_3, 2.0)))
	else:
		tmp = 2.0 / math.pow((math.sqrt(2.0) * (k * t_3)), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 2.4e-89)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 4.2e-39)
		tmp = Float64(2.0 / Float64(Float64(1.0 / l) * Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	elseif (t_m <= 2.1e+167)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * (t_3 ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * t_3)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	t_3 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (t_m <= 2.4e-89)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / ((l ^ 2.0) * cos(k)));
	elseif (t_m <= 4.2e-39)
		tmp = 2.0 / ((1.0 / l) * ((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m ^ 3.0) / l))));
	elseif (t_m <= 2.1e+167)
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * (t_3 ^ 2.0)));
	else
		tmp = 2.0 / ((sqrt(2.0) * (k * t_3)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.4e-89], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e-39], N[(2.0 / N[(N[(1.0 / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+167], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.40000000000000016e-89

   1. Initial program 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 62.3%

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

      1. associate-*r* 62.4%

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

   7. Simplified 62.4%

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

   if 2.40000000000000016e-89 < t < 4.19999999999999987e-39

   1. Initial program 54.8%

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

   3. Simplified 54.7%

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

      1. associate-*l* 54.6%

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

      2. associate-/r* 67.3%

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

      3. associate-+r+ 67.3%

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

      4. metadata-eval 67.3%

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

      5. associate-*l* 67.4%

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

      6. associate-*l/ 68.2%

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

      7. clear-num 68.2%

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

      8. associate-*l* 68.2%

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

   6. Applied egg-rr 68.2%

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

      1. associate-/r/ 68.2%

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

      2. associate-*r* 68.1%

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

   8. Simplified 68.1%

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

   if 4.19999999999999987e-39 < t < 2.0999999999999999e167

   1. Initial program 68.1%

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

   3. Simplified 68.0%

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

      1. add-sqr-sqrt 68.0%

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

      2. pow2 68.0%

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

      3. sqrt-div 68.1%

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

      4. sqrt-pow1 68.9%

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

      5. metadata-eval 68.9%

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

      6. sqrt-prod 57.1%

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

      7. add-sqr-sqrt 87.8%

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

   6. Applied egg-rr 87.8%

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

   if 2.0999999999999999e167 < t

   1. Initial program 68.1%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in k around 0 65.8%

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

      1. add-sqr-sqrt 65.8%

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

      2. pow2 65.8%

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

      3. associate-*r* 65.8%

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

      4. sqrt-prod 65.8%

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

      5. associate-/l/ 61.5%

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

      6. unpow2 61.5%

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

      7. div-inv 61.5%

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

      8. pow-flip 61.5%

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

      9. metadata-eval 61.5%

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

      10. sqrt-pow1 68.1%

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

      11. metadata-eval 68.1%

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

      12. pow1 68.1%

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

   7. Applied egg-rr 68.1%

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

      1. pow1 68.1%

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

      2. *-commutative 68.1%

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

      3. sqrt-prod 68.1%

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

      4. sqrt-prod 68.1%

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

      5. sqrt-pow1 71.3%

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

      6. metadata-eval 71.3%

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

      7. sqrt-pow1 75.7%

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

      8. metadata-eval 75.7%

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

      9. unpow-1 75.7%

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

   9. Applied egg-rr 75.7%

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

       1. unpow1 75.7%

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

       2. *-commutative 75.7%

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

       3. *-commutative 75.7%

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

       4. associate-*l* 75.7%

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

       5. associate-*r/ 75.7%

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

       6. *-rgt-identity 75.7%

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

   11. Simplified 75.7%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{1}{\ell} \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+167}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t\_m \cdot k}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-28)
    (/ 2.0 (pow (* (sqrt 2.0) (* k (/ (pow t_m 1.5) l))) 2.0))
    (if (<= k 2e+152)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (/ 2.0 (pow (* (* k (cbrt (pow l -2.0))) (cbrt (* t_m k))) 3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-28) {
		tmp = 2.0 / pow((sqrt(2.0) * (k * (pow(t_m, 1.5) / l))), 2.0);
	} else if (k <= 2e+152) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / pow(((k * cbrt(pow(l, -2.0))) * cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-28) {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	} else if (k <= 2e+152) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else {
		tmp = 2.0 / Math.pow(((k * Math.cbrt(Math.pow(l, -2.0))) * Math.cbrt((t_m * k))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.7e-28)
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	elseif (k <= 2e+152)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * cbrt((l ^ -2.0))) * cbrt(Float64(t_m * k))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.7e-28], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e+152], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.7e-28

   1. Initial program 54.0%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in k around 0 55.8%

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

      1. add-sqr-sqrt 28.4%

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

      2. pow2 28.4%

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

      3. associate-*r* 28.4%

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

      4. sqrt-prod 28.4%

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

      5. associate-/l/ 27.5%

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

      6. unpow2 27.5%

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

      7. div-inv 27.5%

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

      8. pow-flip 27.5%

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

      9. metadata-eval 27.5%

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

      10. sqrt-pow1 29.4%

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

      11. metadata-eval 29.4%

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

      12. pow1 29.4%

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

   7. Applied egg-rr 29.4%

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

      1. pow1 29.4%

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

      2. *-commutative 29.4%

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

      3. sqrt-prod 29.4%

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

      4. sqrt-prod 30.0%

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

      5. sqrt-pow1 28.8%

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

      6. metadata-eval 28.8%

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

      7. sqrt-pow1 31.7%

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

      8. metadata-eval 31.7%

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

      9. unpow-1 31.7%

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

   9. Applied egg-rr 31.7%

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

       1. unpow1 31.7%

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

       2. *-commutative 31.7%

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

       3. *-commutative 31.7%

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

       4. associate-*l* 31.7%

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

       5. associate-*r/ 31.7%

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

       6. *-rgt-identity 31.7%

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

   11. Simplified 31.7%

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

   if 1.7e-28 < k < 2.0000000000000001e152

   1. Initial program 39.6%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in t around 0 71.3%

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

      1. associate-*r* 71.3%

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

   7. Simplified 71.3%

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

   if 2.0000000000000001e152 < k

   1. Initial program 37.1%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0 37.4%

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

      1. associate-/l* 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in k around inf 56.7%

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

      1. times-frac 56.7%

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

   10. Simplified 56.7%

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

       1. add-cube-cbrt 56.7%

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

       2. pow3 56.7%

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

       3. cbrt-prod 56.7%

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

       4. div-inv 56.7%

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

       5. cbrt-prod 56.7%

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

       6. unpow3 56.7%

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

       7. add-cbrt-cube 59.6%

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

       8. pow-flip 59.6%

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

       9. metadata-eval 59.6%

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

       10. associate-/l* 59.6%

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

       11. tan-quot 59.6%

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

   12. Applied egg-rr 59.6%

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

   13. Taylor expanded in k around 0 59.7%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \sqrt[3]{{\ell}^{-2}}\right) \cdot \sqrt[3]{t \cdot k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t\_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + t\_2\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.4e-121)
      (/ 2.0 (pow (* (pow k 2.0) (/ (sqrt t_m) l)) 2.0))
      (if (<= t_m 6.8e-22)
        (/
         2.0
         (*
          (* (tan k) (+ 1.0 (+ t_2 1.0)))
          (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
        (if (<= t_m 5.3e+102)
          (/
           2.0
           (/ (* (* (tan k) (+ 2.0 t_2)) (* (sin k) (/ (pow t_m 3.0) l))) l))
          (/ 2.0 (pow (* (sqrt 2.0) (* k (/ (pow t_m 1.5) l))) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.4e-121) {
		tmp = 2.0 / pow((pow(k, 2.0) * (sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 6.8e-22) {
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else if (t_m <= 5.3e+102) {
		tmp = 2.0 / (((tan(k) * (2.0 + t_2)) * (sin(k) * (pow(t_m, 3.0) / l))) / l);
	} else {
		tmp = 2.0 / pow((sqrt(2.0) * (k * (pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 1.4d-121) then
        tmp = 2.0d0 / (((k ** 2.0d0) * (sqrt(t_m) / l)) ** 2.0d0)
    else if (t_m <= 6.8d-22) then
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (t_2 + 1.0d0))) * (sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))))
    else if (t_m <= 5.3d+102) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + t_2)) * (sin(k) * ((t_m ** 3.0d0) / l))) / l)
    else
        tmp = 2.0d0 / ((sqrt(2.0d0) * (k * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.4e-121) {
		tmp = 2.0 / Math.pow((Math.pow(k, 2.0) * (Math.sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 6.8e-22) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (t_2 + 1.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else if (t_m <= 5.3e+102) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))) / l);
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 1.4e-121:
		tmp = 2.0 / math.pow((math.pow(k, 2.0) * (math.sqrt(t_m) / l)), 2.0)
	elif t_m <= 6.8e-22:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (t_2 + 1.0))) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))))
	elif t_m <= 5.3e+102:
		tmp = 2.0 / (((math.tan(k) * (2.0 + t_2)) * (math.sin(k) * (math.pow(t_m, 3.0) / l))) / l)
	else:
		tmp = 2.0 / math.pow((math.sqrt(2.0) * (k * (math.pow(t_m, 1.5) / l))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.4e-121)
		tmp = Float64(2.0 / (Float64((k ^ 2.0) * Float64(sqrt(t_m) / l)) ^ 2.0));
	elseif (t_m <= 6.8e-22)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	elseif (t_m <= 5.3e+102)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l))) / l));
	else
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.4e-121)
		tmp = 2.0 / (((k ^ 2.0) * (sqrt(t_m) / l)) ^ 2.0);
	elseif (t_m <= 6.8e-22)
		tmp = 2.0 / ((tan(k) * (1.0 + (t_2 + 1.0))) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))));
	elseif (t_m <= 5.3e+102)
		tmp = 2.0 / (((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m ^ 3.0) / l))) / l);
	else
		tmp = 2.0 / ((sqrt(2.0) * (k * ((t_m ^ 1.5) / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.4e-121], N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.8e-22], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.3e+102], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.4000000000000001e-121

   1. Initial program 42.2%

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

   3. Simplified 42.2%

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

   5. Taylor expanded in k around 0 40.9%

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

      1. associate-/l* 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in k around inf 53.1%

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

      1. times-frac 53.1%

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

   10. Simplified 53.1%

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

   11. Taylor expanded in k around 0 52.4%

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

       1. associate-/l* 52.3%

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

   13. Simplified 52.3%

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

       1. add-sqr-sqrt 12.5%

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

       2. pow2 12.5%

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

       3. sqrt-prod 12.5%

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

       4. sqrt-pow1 13.7%

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

       5. metadata-eval 13.7%

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

       6. sqrt-div 11.4%

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

       7. sqrt-pow1 12.5%

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

       8. metadata-eval 12.5%

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

       9. pow1 12.5%

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

   15. Applied egg-rr 12.5%

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

   if 1.4000000000000001e-121 < t < 6.7999999999999997e-22

   1. Initial program 48.3%

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

   3. Simplified 48.3%

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

      1. unpow3 48.3%

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

      2. times-frac 62.9%

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

      3. pow2 62.9%

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

   6. Applied egg-rr 62.9%

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

   if 6.7999999999999997e-22 < t < 5.2999999999999997e102

   1. Initial program 76.6%

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

   3. Simplified 76.5%

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

      1. associate-*l* 76.4%

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

      2. associate-/r* 81.1%

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

      3. associate-+r+ 81.1%

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

      4. metadata-eval 81.1%

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

      5. associate-*l* 81.1%

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

      6. associate-*l/ 85.8%

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

      7. associate-*l* 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. associate-*r* 85.9%

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

   8. Simplified 85.9%

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

   if 5.2999999999999997e102 < t

   1. Initial program 62.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in k around 0 58.8%

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

      1. add-sqr-sqrt 58.8%

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

      2. pow2 58.8%

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

      3. associate-*r* 58.8%

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

      4. sqrt-prod 58.8%

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

      5. associate-/l/ 55.1%

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

      6. unpow2 55.1%

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

      7. div-inv 55.1%

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

      8. pow-flip 55.1%

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

      9. metadata-eval 55.1%

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

      10. sqrt-pow1 62.8%

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

      11. metadata-eval 62.8%

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

      12. pow1 62.8%

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

   7. Applied egg-rr 62.8%

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

      1. pow1 62.8%

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

      2. *-commutative 62.8%

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

      3. sqrt-prod 62.8%

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

      4. sqrt-prod 62.8%

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

      5. sqrt-pow1 65.8%

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

      6. metadata-eval 65.8%

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

      7. sqrt-pow1 73.8%

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

      8. metadata-eval 73.8%

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

      9. unpow-1 73.8%

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

   9. Applied egg-rr 73.8%

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

       1. unpow1 73.8%

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

       2. *-commutative 73.8%

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

       3. *-commutative 73.8%

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

       4. associate-*l* 73.8%

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

       5. associate-*r/ 73.8%

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

       6. *-rgt-identity 73.8%

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

   11. Simplified 73.8%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.66 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{{t\_m}^{3}}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{2 \cdot \ell}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.1e-95)
    (/ 2.0 (pow (* (pow k 2.0) (/ (sqrt t_m) l)) 2.0))
    (if (<= t_m 1.66e-46)
      (/ 2.0 (pow (* k (* (/ (sqrt 2.0) l) (sqrt (pow t_m 3.0)))) 2.0))
      (if (<= t_m 3.4e+31)
        (*
         (/ (* 2.0 l) (* (pow t_m 3.0) (* (sin k) (tan k))))
         (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
        (/ 2.0 (pow (* (sqrt 2.0) (* k (/ (pow t_m 1.5) l))) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.1e-95) {
		tmp = 2.0 / pow((pow(k, 2.0) * (sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 1.66e-46) {
		tmp = 2.0 / pow((k * ((sqrt(2.0) / l) * sqrt(pow(t_m, 3.0)))), 2.0);
	} else if (t_m <= 3.4e+31) {
		tmp = ((2.0 * l) / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / pow((sqrt(2.0) * (k * (pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.1d-95) then
        tmp = 2.0d0 / (((k ** 2.0d0) * (sqrt(t_m) / l)) ** 2.0d0)
    else if (t_m <= 1.66d-46) then
        tmp = 2.0d0 / ((k * ((sqrt(2.0d0) / l) * sqrt((t_m ** 3.0d0)))) ** 2.0d0)
    else if (t_m <= 3.4d+31) then
        tmp = ((2.0d0 * l) / ((t_m ** 3.0d0) * (sin(k) * tan(k)))) * (l / (2.0d0 + ((k / t_m) ** 2.0d0)))
    else
        tmp = 2.0d0 / ((sqrt(2.0d0) * (k * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.1e-95) {
		tmp = 2.0 / Math.pow((Math.pow(k, 2.0) * (Math.sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 1.66e-46) {
		tmp = 2.0 / Math.pow((k * ((Math.sqrt(2.0) / l) * Math.sqrt(Math.pow(t_m, 3.0)))), 2.0);
	} else if (t_m <= 3.4e+31) {
		tmp = ((2.0 * l) / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.1e-95:
		tmp = 2.0 / math.pow((math.pow(k, 2.0) * (math.sqrt(t_m) / l)), 2.0)
	elif t_m <= 1.66e-46:
		tmp = 2.0 / math.pow((k * ((math.sqrt(2.0) / l) * math.sqrt(math.pow(t_m, 3.0)))), 2.0)
	elif t_m <= 3.4e+31:
		tmp = ((2.0 * l) / (math.pow(t_m, 3.0) * (math.sin(k) * math.tan(k)))) * (l / (2.0 + math.pow((k / t_m), 2.0)))
	else:
		tmp = 2.0 / math.pow((math.sqrt(2.0) * (k * (math.pow(t_m, 1.5) / l))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.1e-95)
		tmp = Float64(2.0 / (Float64((k ^ 2.0) * Float64(sqrt(t_m) / l)) ^ 2.0));
	elseif (t_m <= 1.66e-46)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(sqrt(2.0) / l) * sqrt((t_m ^ 3.0)))) ^ 2.0));
	elseif (t_m <= 3.4e+31)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.1e-95)
		tmp = 2.0 / (((k ^ 2.0) * (sqrt(t_m) / l)) ^ 2.0);
	elseif (t_m <= 1.66e-46)
		tmp = 2.0 / ((k * ((sqrt(2.0) / l) * sqrt((t_m ^ 3.0)))) ^ 2.0);
	elseif (t_m <= 3.4e+31)
		tmp = ((2.0 * l) / ((t_m ^ 3.0) * (sin(k) * tan(k)))) * (l / (2.0 + ((k / t_m) ^ 2.0)));
	else
		tmp = 2.0 / ((sqrt(2.0) * (k * ((t_m ^ 1.5) / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.1e-95], N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.66e-46], N[(2.0 / N[Power[N[(k * N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+31], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 4.0999999999999997e-95

   1. Initial program 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 40.0%

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

      1. associate-/l* 40.2%

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

   7. Simplified 40.2%

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

   8. Taylor expanded in k around inf 52.5%

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

      1. times-frac 52.6%

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

   10. Simplified 52.6%

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

   11. Taylor expanded in k around 0 51.9%

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

       1. associate-/l* 51.7%

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

   13. Simplified 51.7%

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

       1. add-sqr-sqrt 12.9%

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

       2. pow2 12.9%

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

       3. sqrt-prod 12.8%

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

       4. sqrt-pow1 14.0%

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

       5. metadata-eval 14.0%

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

       6. sqrt-div 11.8%

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

       7. sqrt-pow1 12.9%

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

       8. metadata-eval 12.9%

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

       9. pow1 12.9%

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

   15. Applied egg-rr 12.9%

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

   if 4.0999999999999997e-95 < t < 1.6599999999999999e-46

   1. Initial program 54.8%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in k around 0 50.2%

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

      1. add-sqr-sqrt 50.2%

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

      2. pow2 50.2%

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

      3. associate-*r* 50.2%

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

      4. sqrt-prod 50.2%

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

      5. associate-/l/ 50.4%

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

      6. unpow2 50.4%

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

      7. div-inv 50.4%

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

      8. pow-flip 50.4%

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

      9. metadata-eval 50.4%

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

      10. sqrt-pow1 50.0%

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

      11. metadata-eval 50.0%

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

      12. pow1 50.0%

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

   7. Applied egg-rr 50.0%

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

   8. Taylor expanded in t around 0 54.4%

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

   if 1.6599999999999999e-46 < t < 3.3999999999999998e31

   1. Initial program 69.5%

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

   3. Simplified 67.7%

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

      1. associate-*r* 74.5%

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

      2. *-un-lft-identity 74.5%

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

      3. times-frac 81.2%

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

      4. associate-*r* 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. /-rgt-identity 81.0%

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

      2. associate-*l/ 80.7%

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

      3. associate-*l* 81.0%

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

   8. Simplified 81.0%

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

   if 3.3999999999999998e31 < t

   1. Initial program 67.0%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in k around 0 63.8%

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

      1. add-sqr-sqrt 63.8%

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

      2. pow2 63.8%

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

      3. associate-*r* 63.8%

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

      4. sqrt-prod 63.8%

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

      5. associate-/l/ 60.8%

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

      6. unpow2 60.8%

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

      7. div-inv 60.8%

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

      8. pow-flip 60.8%

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

      9. metadata-eval 60.8%

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

      10. sqrt-pow1 66.9%

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

      11. metadata-eval 66.9%

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

      12. pow1 66.9%

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

   7. Applied egg-rr 66.9%

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

      1. pow1 66.9%

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

      2. *-commutative 66.9%

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

      3. sqrt-prod 66.9%

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

      4. sqrt-prod 68.8%

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

      5. sqrt-pow1 71.2%

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

      6. metadata-eval 71.2%

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

      7. sqrt-pow1 77.5%

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

      8. metadata-eval 77.5%

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

      9. unpow-1 77.5%

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

   9. Applied egg-rr 77.5%

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

       1. unpow1 77.5%

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

       2. *-commutative 77.5%

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

       3. *-commutative 77.5%

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

       4. associate-*l* 77.5%

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

       5. associate-*r/ 77.5%

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

       6. *-rgt-identity 77.5%

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

   11. Simplified 77.5%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{\sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-113)
    (/ 2.0 (pow (* (pow k 2.0) (/ (sqrt t_m) l)) 2.0))
    (if (<= t_m 7.5e+29)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
         (* (sin k) (/ (pow t_m 3.0) l)))
        l))
      (/ 2.0 (pow (* (sqrt 2.0) (* k (/ (pow t_m 1.5) l))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-113) {
		tmp = 2.0 / pow((pow(k, 2.0) * (sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 7.5e+29) {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * (pow(t_m, 3.0) / l))) / l);
	} else {
		tmp = 2.0 / pow((sqrt(2.0) * (k * (pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.4d-113) then
        tmp = 2.0d0 / (((k ** 2.0d0) * (sqrt(t_m) / l)) ** 2.0d0)
    else if (t_m <= 7.5d+29) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * (sin(k) * ((t_m ** 3.0d0) / l))) / l)
    else
        tmp = 2.0d0 / ((sqrt(2.0d0) * (k * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-113) {
		tmp = 2.0 / Math.pow((Math.pow(k, 2.0) * (Math.sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 7.5e+29) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l))) / l);
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.4e-113:
		tmp = 2.0 / math.pow((math.pow(k, 2.0) * (math.sqrt(t_m) / l)), 2.0)
	elif t_m <= 7.5e+29:
		tmp = 2.0 / (((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * (math.sin(k) * (math.pow(t_m, 3.0) / l))) / l)
	else:
		tmp = 2.0 / math.pow((math.sqrt(2.0) * (k * (math.pow(t_m, 1.5) / l))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.4e-113)
		tmp = Float64(2.0 / (Float64((k ^ 2.0) * Float64(sqrt(t_m) / l)) ^ 2.0));
	elseif (t_m <= 7.5e+29)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64((t_m ^ 3.0) / l))) / l));
	else
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.4e-113)
		tmp = 2.0 / (((k ^ 2.0) * (sqrt(t_m) / l)) ^ 2.0);
	elseif (t_m <= 7.5e+29)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * (sin(k) * ((t_m ^ 3.0) / l))) / l);
	else
		tmp = 2.0 / ((sqrt(2.0) * (k * ((t_m ^ 1.5) / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-113], N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.5e+29], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.4000000000000002e-113

   1. Initial program 41.7%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in k around 0 40.4%

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

      1. associate-/l* 40.7%

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

   7. Simplified 40.7%

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

   8. Taylor expanded in k around inf 53.1%

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

      1. times-frac 53.1%

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

   10. Simplified 53.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 52.3%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   13. Simplified 52.3%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 13.0%

          \[\leadsto \frac{2}{\color{blue}{\sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}} \cdot \sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}}} \]

       2. pow2 13.0%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\right)}^{2}}} \]

       3. sqrt-prod 13.0%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}}^{2}} \]

       4. sqrt-pow1 14.2%

          \[\leadsto \frac{2}{{\left(\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}^{2}} \]

       5. metadata-eval 14.2%

          \[\leadsto \frac{2}{{\left({k}^{\color{blue}{2}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}^{2}} \]

       6. sqrt-div 11.9%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \color{blue}{\frac{\sqrt{t}}{\sqrt{{\ell}^{2}}}}\right)}^{2}} \]

       7. sqrt-pow1 13.0%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}\right)}^{2}} \]

       8. metadata-eval 13.0%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{{\ell}^{\color{blue}{1}}}\right)}^{2}} \]

       9. pow1 13.0%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\color{blue}{\ell}}\right)}^{2}} \]

   15. Applied egg-rr 13.0%

       \[\leadsto \frac{2}{\color{blue}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}} \]

   if 3.4000000000000002e-113 < t < 7.49999999999999945e29

   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 58.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* 57.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]

      2. associate-/r* 63.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]

      3. associate-+r+ 63.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      4. metadata-eval 63.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      5. associate-*l* 63.4%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      6. associate-*l/ 66.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

      7. associate-*l* 66.9%

         \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}{\ell}} \]

   6. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{\ell}}} \]
   7. Step-by-step derivation

      1. associate-*r* 66.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\ell}} \]

   8. Simplified 66.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}} \]

   if 7.49999999999999945e29 < t

   1. Initial program 67.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 63.8%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}}} \]

      2. pow2 63.8%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\right)}^{2}}} \]

      3. associate-*r* 63.8%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2\right) \cdot {k}^{2}}}\right)}^{2}} \]

      4. sqrt-prod 63.8%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}}^{2}} \]

      5. associate-/l/ 60.8%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      6. unpow2 60.8%

         \[\leadsto \frac{2}{{\left(\sqrt{\frac{{t}^{3}}{\color{blue}{{\ell}^{2}}} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      7. div-inv 60.8%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left({t}^{3} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      8. pow-flip 60.8%

         \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      9. metadata-eval 60.8%

         \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      10. sqrt-pow1 66.9%

          \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot \color{blue}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{2}} \]

      11. metadata-eval 66.9%

          \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot {k}^{\color{blue}{1}}\right)}^{2}} \]

      12. pow1 66.9%

          \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot \color{blue}{k}\right)}^{2}} \]

   7. Applied egg-rr 66.9%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot k\right)}^{2}}} \]
   8. Step-by-step derivation

      1. pow1 66.9%

         \[\leadsto \frac{2}{{\color{blue}{\left({\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot k\right)}^{1}\right)}}^{2}} \]

      2. *-commutative 66.9%

         \[\leadsto \frac{2}{{\left({\color{blue}{\left(k \cdot \sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2}\right)}}^{1}\right)}^{2}} \]

      3. sqrt-prod 66.9%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \color{blue}{\left(\sqrt{{t}^{3} \cdot {\ell}^{-2}} \cdot \sqrt{2}\right)}\right)}^{1}\right)}^{2}} \]

      4. sqrt-prod 68.8%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\color{blue}{\left(\sqrt{{t}^{3}} \cdot \sqrt{{\ell}^{-2}}\right)} \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      5. sqrt-pow1 71.2%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left(\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{{\ell}^{-2}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      6. metadata-eval 71.2%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{\color{blue}{1.5}} \cdot \sqrt{{\ell}^{-2}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      7. sqrt-pow1 77.5%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot \color{blue}{{\ell}^{\left(\frac{-2}{2}\right)}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      8. metadata-eval 77.5%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot {\ell}^{\color{blue}{-1}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      9. unpow-1 77.5%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot \color{blue}{\frac{1}{\ell}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

   9. Applied egg-rr 77.5%

      \[\leadsto \frac{2}{{\color{blue}{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}}^{2}} \]
   10. Step-by-step derivation

       1. unpow1 77.5%

          \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{2}\right)\right)}}^{2}} \]

       2. *-commutative 77.5%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{2}\right) \cdot k\right)}}^{2}} \]

       3. *-commutative 77.5%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{2} \cdot \left({t}^{1.5} \cdot \frac{1}{\ell}\right)\right)} \cdot k\right)}^{2}} \]

       4. associate-*l* 77.5%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{2} \cdot \left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot k\right)\right)}}^{2}} \]

       5. associate-*r/ 77.5%

          \[\leadsto \frac{2}{{\left(\sqrt{2} \cdot \left(\color{blue}{\frac{{t}^{1.5} \cdot 1}{\ell}} \cdot k\right)\right)}^{2}} \]

       6. *-rgt-identity 77.5%

          \[\leadsto \frac{2}{{\left(\sqrt{2} \cdot \left(\frac{\color{blue}{{t}^{1.5}}}{\ell} \cdot k\right)\right)}^{2}} \]

   11. Simplified 77.5%

       \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot k\right)\right)}}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t\_m}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-124)
    (/ 2.0 (pow (* (pow k 2.0) (/ (sqrt t_m) l)) 2.0))
    (/ 2.0 (pow (* (sqrt 2.0) (* k (/ (pow t_m 1.5) l))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-124) {
		tmp = 2.0 / pow((pow(k, 2.0) * (sqrt(t_m) / l)), 2.0);
	} else {
		tmp = 2.0 / pow((sqrt(2.0) * (k * (pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.5d-124) then
        tmp = 2.0d0 / (((k ** 2.0d0) * (sqrt(t_m) / l)) ** 2.0d0)
    else
        tmp = 2.0d0 / ((sqrt(2.0d0) * (k * ((t_m ** 1.5d0) / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-124) {
		tmp = 2.0 / Math.pow((Math.pow(k, 2.0) * (Math.sqrt(t_m) / l)), 2.0);
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt(2.0) * (k * (Math.pow(t_m, 1.5) / l))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.5e-124:
		tmp = 2.0 / math.pow((math.pow(k, 2.0) * (math.sqrt(t_m) / l)), 2.0)
	else:
		tmp = 2.0 / math.pow((math.sqrt(2.0) * (k * (math.pow(t_m, 1.5) / l))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-124)
		tmp = Float64(2.0 / (Float64((k ^ 2.0) * Float64(sqrt(t_m) / l)) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(sqrt(2.0) * Float64(k * Float64((t_m ^ 1.5) / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.5e-124)
		tmp = 2.0 / (((k ^ 2.0) * (sqrt(t_m) / l)) ^ 2.0);
	else
		tmp = 2.0 / ((sqrt(2.0) * (k * ((t_m ^ 1.5) / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-124], N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999996e-124

   1. Initial program 42.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 41.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 41.2%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 53.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 53.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 53.1%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 52.3%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   13. Simplified 52.3%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 12.5%

          \[\leadsto \frac{2}{\color{blue}{\sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}} \cdot \sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}}} \]

       2. pow2 12.5%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\right)}^{2}}} \]

       3. sqrt-prod 12.5%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}}^{2}} \]

       4. sqrt-pow1 13.7%

          \[\leadsto \frac{2}{{\left(\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}^{2}} \]

       5. metadata-eval 13.7%

          \[\leadsto \frac{2}{{\left({k}^{\color{blue}{2}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}^{2}} \]

       6. sqrt-div 11.4%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \color{blue}{\frac{\sqrt{t}}{\sqrt{{\ell}^{2}}}}\right)}^{2}} \]

       7. sqrt-pow1 12.5%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}\right)}^{2}} \]

       8. metadata-eval 12.5%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{{\ell}^{\color{blue}{1}}}\right)}^{2}} \]

       9. pow1 12.5%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\color{blue}{\ell}}\right)}^{2}} \]

   15. Applied egg-rr 12.5%

       \[\leadsto \frac{2}{\color{blue}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}} \]

   if 4.4999999999999996e-124 < t

   1. Initial program 62.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 58.4%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 58.4%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}}} \]

      2. pow2 58.4%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}\right)}^{2}}} \]

      3. associate-*r* 58.4%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2\right) \cdot {k}^{2}}}\right)}^{2}} \]

      4. sqrt-prod 58.4%

         \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}}^{2}} \]

      5. associate-/l/ 55.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      6. unpow2 55.5%

         \[\leadsto \frac{2}{{\left(\sqrt{\frac{{t}^{3}}{\color{blue}{{\ell}^{2}}} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      7. div-inv 55.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\left({t}^{3} \cdot \frac{1}{{\ell}^{2}}\right)} \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      8. pow-flip 55.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right) \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      9. metadata-eval 55.3%

         \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{\color{blue}{-2}}\right) \cdot 2} \cdot \sqrt{{k}^{2}}\right)}^{2}} \]

      10. sqrt-pow1 59.0%

          \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot \color{blue}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{2}} \]

      11. metadata-eval 59.0%

          \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot {k}^{\color{blue}{1}}\right)}^{2}} \]

      12. pow1 59.0%

          \[\leadsto \frac{2}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot \color{blue}{k}\right)}^{2}} \]

   7. Applied egg-rr 59.0%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot k\right)}^{2}}} \]
   8. Step-by-step derivation

      1. pow1 59.0%

         \[\leadsto \frac{2}{{\color{blue}{\left({\left(\sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2} \cdot k\right)}^{1}\right)}}^{2}} \]

      2. *-commutative 59.0%

         \[\leadsto \frac{2}{{\left({\color{blue}{\left(k \cdot \sqrt{\left({t}^{3} \cdot {\ell}^{-2}\right) \cdot 2}\right)}}^{1}\right)}^{2}} \]

      3. sqrt-prod 59.0%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \color{blue}{\left(\sqrt{{t}^{3} \cdot {\ell}^{-2}} \cdot \sqrt{2}\right)}\right)}^{1}\right)}^{2}} \]

      4. sqrt-prod 60.2%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\color{blue}{\left(\sqrt{{t}^{3}} \cdot \sqrt{{\ell}^{-2}}\right)} \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      5. sqrt-pow1 62.8%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left(\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \sqrt{{\ell}^{-2}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      6. metadata-eval 62.8%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{\color{blue}{1.5}} \cdot \sqrt{{\ell}^{-2}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      7. sqrt-pow1 69.8%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot \color{blue}{{\ell}^{\left(\frac{-2}{2}\right)}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      8. metadata-eval 69.8%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot {\ell}^{\color{blue}{-1}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

      9. unpow-1 69.8%

         \[\leadsto \frac{2}{{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot \color{blue}{\frac{1}{\ell}}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}^{2}} \]

   9. Applied egg-rr 69.8%

      \[\leadsto \frac{2}{{\color{blue}{\left({\left(k \cdot \left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{2}\right)\right)}^{1}\right)}}^{2}} \]
   10. Step-by-step derivation

       1. unpow1 69.8%

          \[\leadsto \frac{2}{{\color{blue}{\left(k \cdot \left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{2}\right)\right)}}^{2}} \]

       2. *-commutative 69.8%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot \sqrt{2}\right) \cdot k\right)}}^{2}} \]

       3. *-commutative 69.8%

          \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{2} \cdot \left({t}^{1.5} \cdot \frac{1}{\ell}\right)\right)} \cdot k\right)}^{2}} \]

       4. associate-*l* 69.8%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{2} \cdot \left(\left({t}^{1.5} \cdot \frac{1}{\ell}\right) \cdot k\right)\right)}}^{2}} \]

       5. associate-*r/ 69.8%

          \[\leadsto \frac{2}{{\left(\sqrt{2} \cdot \left(\color{blue}{\frac{{t}^{1.5} \cdot 1}{\ell}} \cdot k\right)\right)}^{2}} \]

       6. *-rgt-identity 69.8%

          \[\leadsto \frac{2}{{\left(\sqrt{2} \cdot \left(\frac{\color{blue}{{t}^{1.5}}}{\ell} \cdot k\right)\right)}^{2}} \]

   11. Simplified 69.8%

       \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot k\right)\right)}}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{2} \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t\_m}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k 2.0))))
   (*
    t_s
    (if (<= t_m 3.1e-106)
      (/ 2.0 (pow (* (pow k 2.0) (/ (sqrt t_m) l)) 2.0))
      (if (<= t_m 2e+31)
        (/ 2.0 (* (/ t_2 l) (/ (pow t_m 3.0) l)))
        (/ 2.0 (* t_2 (/ (* t_m (/ (pow t_m 2.0) l)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * pow(k, 2.0);
	double tmp;
	if (t_m <= 3.1e-106) {
		tmp = 2.0 / pow((pow(k, 2.0) * (sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 2e+31) {
		tmp = 2.0 / ((t_2 / l) * (pow(t_m, 3.0) / l));
	} else {
		tmp = 2.0 / (t_2 * ((t_m * (pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (k ** 2.0d0)
    if (t_m <= 3.1d-106) then
        tmp = 2.0d0 / (((k ** 2.0d0) * (sqrt(t_m) / l)) ** 2.0d0)
    else if (t_m <= 2d+31) then
        tmp = 2.0d0 / ((t_2 / l) * ((t_m ** 3.0d0) / l))
    else
        tmp = 2.0d0 / (t_2 * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * Math.pow(k, 2.0);
	double tmp;
	if (t_m <= 3.1e-106) {
		tmp = 2.0 / Math.pow((Math.pow(k, 2.0) * (Math.sqrt(t_m) / l)), 2.0);
	} else if (t_m <= 2e+31) {
		tmp = 2.0 / ((t_2 / l) * (Math.pow(t_m, 3.0) / l));
	} else {
		tmp = 2.0 / (t_2 * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 * math.pow(k, 2.0)
	tmp = 0
	if t_m <= 3.1e-106:
		tmp = 2.0 / math.pow((math.pow(k, 2.0) * (math.sqrt(t_m) / l)), 2.0)
	elif t_m <= 2e+31:
		tmp = 2.0 / ((t_2 / l) * (math.pow(t_m, 3.0) / l))
	else:
		tmp = 2.0 / (t_2 * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 * (k ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.1e-106)
		tmp = Float64(2.0 / (Float64((k ^ 2.0) * Float64(sqrt(t_m) / l)) ^ 2.0));
	elseif (t_m <= 2e+31)
		tmp = Float64(2.0 / Float64(Float64(t_2 / l) * Float64((t_m ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 * (k ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.1e-106)
		tmp = 2.0 / (((k ^ 2.0) * (sqrt(t_m) / l)) ^ 2.0);
	elseif (t_m <= 2e+31)
		tmp = 2.0 / ((t_2 / l) * ((t_m ^ 3.0) / l));
	else
		tmp = 2.0 / (t_2 * ((t_m * ((t_m ^ 2.0) / l)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-106], N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+31], N[(2.0 / N[(N[(t$95$2 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.09999999999999985e-106

   1. Initial program 41.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 41.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.2%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 51.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. associate-/l* 51.7%

          \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   13. Simplified 51.7%

       \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 12.9%

          \[\leadsto \frac{2}{\color{blue}{\sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}} \cdot \sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}}} \]

       2. pow2 12.9%

          \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\right)}^{2}}} \]

       3. sqrt-prod 12.8%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}}^{2}} \]

       4. sqrt-pow1 14.0%

          \[\leadsto \frac{2}{{\left(\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}^{2}} \]

       5. metadata-eval 14.0%

          \[\leadsto \frac{2}{{\left({k}^{\color{blue}{2}} \cdot \sqrt{\frac{t}{{\ell}^{2}}}\right)}^{2}} \]

       6. sqrt-div 11.8%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \color{blue}{\frac{\sqrt{t}}{\sqrt{{\ell}^{2}}}}\right)}^{2}} \]

       7. sqrt-pow1 12.9%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}\right)}^{2}} \]

       8. metadata-eval 12.9%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{{\ell}^{\color{blue}{1}}}\right)}^{2}} \]

       9. pow1 12.9%

          \[\leadsto \frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\color{blue}{\ell}}\right)}^{2}} \]

   15. Applied egg-rr 12.9%

       \[\leadsto \frac{2}{\color{blue}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}} \]

   if 3.09999999999999985e-106 < t < 1.9999999999999999e31

   1. Initial program 62.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.8%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 58.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 58.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 59.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   if 1.9999999999999999e31 < t

   1. Initial program 67.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 63.8%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. cube-mult 63.8%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      2. *-un-lft-identity 63.8%

         \[\leadsto \frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      3. times-frac 65.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      4. pow2 65.9%

         \[\leadsto \frac{2}{\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

   7. Applied egg-rr 65.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{\left({k}^{2} \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t\_m} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k 2.0))))
   (*
    t_s
    (if (<= t_m 3.5e-106)
      (/ 2.0 (pow (* (sqrt t_m) (/ (pow k 2.0) l)) 2.0))
      (if (<= t_m 3.4e+31)
        (/ 2.0 (* (/ t_2 l) (/ (pow t_m 3.0) l)))
        (/ 2.0 (* t_2 (/ (* t_m (/ (pow t_m 2.0) l)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * pow(k, 2.0);
	double tmp;
	if (t_m <= 3.5e-106) {
		tmp = 2.0 / pow((sqrt(t_m) * (pow(k, 2.0) / l)), 2.0);
	} else if (t_m <= 3.4e+31) {
		tmp = 2.0 / ((t_2 / l) * (pow(t_m, 3.0) / l));
	} else {
		tmp = 2.0 / (t_2 * ((t_m * (pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (k ** 2.0d0)
    if (t_m <= 3.5d-106) then
        tmp = 2.0d0 / ((sqrt(t_m) * ((k ** 2.0d0) / l)) ** 2.0d0)
    else if (t_m <= 3.4d+31) then
        tmp = 2.0d0 / ((t_2 / l) * ((t_m ** 3.0d0) / l))
    else
        tmp = 2.0d0 / (t_2 * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * Math.pow(k, 2.0);
	double tmp;
	if (t_m <= 3.5e-106) {
		tmp = 2.0 / Math.pow((Math.sqrt(t_m) * (Math.pow(k, 2.0) / l)), 2.0);
	} else if (t_m <= 3.4e+31) {
		tmp = 2.0 / ((t_2 / l) * (Math.pow(t_m, 3.0) / l));
	} else {
		tmp = 2.0 / (t_2 * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 * math.pow(k, 2.0)
	tmp = 0
	if t_m <= 3.5e-106:
		tmp = 2.0 / math.pow((math.sqrt(t_m) * (math.pow(k, 2.0) / l)), 2.0)
	elif t_m <= 3.4e+31:
		tmp = 2.0 / ((t_2 / l) * (math.pow(t_m, 3.0) / l))
	else:
		tmp = 2.0 / (t_2 * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 * (k ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.5e-106)
		tmp = Float64(2.0 / (Float64(sqrt(t_m) * Float64((k ^ 2.0) / l)) ^ 2.0));
	elseif (t_m <= 3.4e+31)
		tmp = Float64(2.0 / Float64(Float64(t_2 / l) * Float64((t_m ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 * (k ^ 2.0);
	tmp = 0.0;
	if (t_m <= 3.5e-106)
		tmp = 2.0 / ((sqrt(t_m) * ((k ^ 2.0) / l)) ^ 2.0);
	elseif (t_m <= 3.4e+31)
		tmp = 2.0 / ((t_2 / l) * ((t_m ^ 3.0) / l));
	else
		tmp = 2.0 / (t_2 * ((t_m * ((t_m ^ 2.0) / l)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-106], N[(2.0 / N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+31], N[(2.0 / N[(N[(t$95$2 / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.5e-106

   1. Initial program 41.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 41.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.2%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 51.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 13.0%

          \[\leadsto \frac{2}{\color{blue}{\sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}}} \]

       2. sqrt-div 11.2%

          \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt{{k}^{4} \cdot t}}{\sqrt{{\ell}^{2}}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       3. sqrt-prod 11.2%

          \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}}{\sqrt{{\ell}^{2}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       4. sqrt-pow1 11.2%

          \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}}{\sqrt{{\ell}^{2}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       5. metadata-eval 11.2%

          \[\leadsto \frac{2}{\frac{{k}^{\color{blue}{2}} \cdot \sqrt{t}}{\sqrt{{\ell}^{2}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       6. sqrt-pow1 6.7%

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       7. metadata-eval 6.7%

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{{\ell}^{\color{blue}{1}}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       8. pow1 6.7%

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\color{blue}{\ell}} \cdot \sqrt{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

       9. sqrt-div 6.7%

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \color{blue}{\frac{\sqrt{{k}^{4} \cdot t}}{\sqrt{{\ell}^{2}}}}} \]

       10. sqrt-prod 6.7%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}}{\sqrt{{\ell}^{2}}}} \]

       11. sqrt-pow1 6.7%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}}{\sqrt{{\ell}^{2}}}} \]

       12. metadata-eval 6.7%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{{k}^{\color{blue}{2}} \cdot \sqrt{t}}{\sqrt{{\ell}^{2}}}} \]

       13. sqrt-pow1 11.9%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{{k}^{2} \cdot \sqrt{t}}{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}} \]

       14. metadata-eval 11.9%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{{k}^{2} \cdot \sqrt{t}}{{\ell}^{\color{blue}{1}}}} \]

       15. pow1 11.9%

           \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{{k}^{2} \cdot \sqrt{t}}{\color{blue}{\ell}}} \]

   13. Applied egg-rr 11.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \sqrt{t}}{\ell} \cdot \frac{{k}^{2} \cdot \sqrt{t}}{\ell}}} \]
   14. Step-by-step derivation

       1. unpow2 11.9%

          \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{k}^{2} \cdot \sqrt{t}}{\ell}\right)}^{2}}} \]

       2. *-commutative 11.9%

          \[\leadsto \frac{2}{{\left(\frac{\color{blue}{\sqrt{t} \cdot {k}^{2}}}{\ell}\right)}^{2}} \]

       3. associate-/l* 12.4%

          \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}}^{2}} \]

   15. Simplified 12.4%

       \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}} \]

   if 3.5e-106 < t < 3.3999999999999998e31

   1. Initial program 62.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 67.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 56.8%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 58.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 58.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 59.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 59.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   if 3.3999999999999998e31 < t

   1. Initial program 67.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 63.8%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. cube-mult 63.8%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      2. *-un-lft-identity 63.8%

         \[\leadsto \frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      3. times-frac 65.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      4. pow2 65.9%

         \[\leadsto \frac{2}{\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

   7. Applied egg-rr 65.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{t} \cdot \frac{{k}^{2}}{\ell}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k}^{2}\\ t_3 := \frac{t\_2}{\ell}\\ t_4 := \frac{{t\_m}^{3}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{t\_3 \cdot t\_4}\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot t\_4}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left({t\_m}^{2} \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k 2.0))) (t_3 (/ t_2 l)) (t_4 (/ (pow t_m 3.0) l)))
   (*
    t_s
    (if (<= t_m 3.1e-106)
      (/ 2.0 (* (* t_m k) (/ (pow k 3.0) (pow l 2.0))))
      (if (<= t_m 2e-39)
        (/ 2.0 (* t_3 t_4))
        (if (<= t_m 1.8e+82)
          (/ 2.0 (/ (* t_2 t_4) l))
          (/ 2.0 (* t_3 (* (pow t_m 2.0) (/ t_m l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * pow(k, 2.0);
	double t_3 = t_2 / l;
	double t_4 = pow(t_m, 3.0) / l;
	double tmp;
	if (t_m <= 3.1e-106) {
		tmp = 2.0 / ((t_m * k) * (pow(k, 3.0) / pow(l, 2.0)));
	} else if (t_m <= 2e-39) {
		tmp = 2.0 / (t_3 * t_4);
	} else if (t_m <= 1.8e+82) {
		tmp = 2.0 / ((t_2 * t_4) / l);
	} else {
		tmp = 2.0 / (t_3 * (pow(t_m, 2.0) * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 * (k ** 2.0d0)
    t_3 = t_2 / l
    t_4 = (t_m ** 3.0d0) / l
    if (t_m <= 3.1d-106) then
        tmp = 2.0d0 / ((t_m * k) * ((k ** 3.0d0) / (l ** 2.0d0)))
    else if (t_m <= 2d-39) then
        tmp = 2.0d0 / (t_3 * t_4)
    else if (t_m <= 1.8d+82) then
        tmp = 2.0d0 / ((t_2 * t_4) / l)
    else
        tmp = 2.0d0 / (t_3 * ((t_m ** 2.0d0) * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 * Math.pow(k, 2.0);
	double t_3 = t_2 / l;
	double t_4 = Math.pow(t_m, 3.0) / l;
	double tmp;
	if (t_m <= 3.1e-106) {
		tmp = 2.0 / ((t_m * k) * (Math.pow(k, 3.0) / Math.pow(l, 2.0)));
	} else if (t_m <= 2e-39) {
		tmp = 2.0 / (t_3 * t_4);
	} else if (t_m <= 1.8e+82) {
		tmp = 2.0 / ((t_2 * t_4) / l);
	} else {
		tmp = 2.0 / (t_3 * (Math.pow(t_m, 2.0) * (t_m / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 * math.pow(k, 2.0)
	t_3 = t_2 / l
	t_4 = math.pow(t_m, 3.0) / l
	tmp = 0
	if t_m <= 3.1e-106:
		tmp = 2.0 / ((t_m * k) * (math.pow(k, 3.0) / math.pow(l, 2.0)))
	elif t_m <= 2e-39:
		tmp = 2.0 / (t_3 * t_4)
	elif t_m <= 1.8e+82:
		tmp = 2.0 / ((t_2 * t_4) / l)
	else:
		tmp = 2.0 / (t_3 * (math.pow(t_m, 2.0) * (t_m / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 * (k ^ 2.0))
	t_3 = Float64(t_2 / l)
	t_4 = Float64((t_m ^ 3.0) / l)
	tmp = 0.0
	if (t_m <= 3.1e-106)
		tmp = Float64(2.0 / Float64(Float64(t_m * k) * Float64((k ^ 3.0) / (l ^ 2.0))));
	elseif (t_m <= 2e-39)
		tmp = Float64(2.0 / Float64(t_3 * t_4));
	elseif (t_m <= 1.8e+82)
		tmp = Float64(2.0 / Float64(Float64(t_2 * t_4) / l));
	else
		tmp = Float64(2.0 / Float64(t_3 * Float64((t_m ^ 2.0) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 * (k ^ 2.0);
	t_3 = t_2 / l;
	t_4 = (t_m ^ 3.0) / l;
	tmp = 0.0;
	if (t_m <= 3.1e-106)
		tmp = 2.0 / ((t_m * k) * ((k ^ 3.0) / (l ^ 2.0)));
	elseif (t_m <= 2e-39)
		tmp = 2.0 / (t_3 * t_4);
	elseif (t_m <= 1.8e+82)
		tmp = 2.0 / ((t_2 * t_4) / l);
	else
		tmp = 2.0 / (t_3 * ((t_m ^ 2.0) * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / l), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-106], N[(2.0 / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e-39], N[(2.0 / N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e+82], N[(2.0 / N[(N[(t$95$2 * t$95$4), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$3 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.09999999999999985e-106

   1. Initial program 41.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 41.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.2%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.5%

       \[\leadsto \frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot t\right)}} \]

   if 3.09999999999999985e-106 < t < 1.99999999999999986e-39

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 65.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 52.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 55.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 55.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 56.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 56.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   if 1.99999999999999986e-39 < t < 1.80000000000000007e82

   1. Initial program 79.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 73.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 73.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 73.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   if 1.80000000000000007e82 < t

   1. Initial program 62.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 58.5%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 58.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 57.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 57.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. unpow3 57.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

       2. *-un-lft-identity 57.9%

          \[\leadsto \frac{2}{\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{1 \cdot \ell}} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

       3. times-frac 58.4%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{1} \cdot \frac{t}{\ell}\right)} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

       4. pow2 58.4%

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{2}}}{1} \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

   11. Applied egg-rr 58.4%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{2}}{1} \cdot \frac{t}{\ell}\right)} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \left({t}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-106)
    (/ 2.0 (* (* t_m k) (/ (pow k 3.0) (pow l 2.0))))
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (* t_m (/ (pow t_m 2.0) l)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-106) {
		tmp = 2.0 / ((t_m * k) * (pow(k, 3.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((t_m * (pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-106) then
        tmp = 2.0d0 / ((t_m * k) * ((k ** 3.0d0) / (l ** 2.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * ((t_m * ((t_m ** 2.0d0) / l)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-106) {
		tmp = 2.0 / ((t_m * k) * (Math.pow(k, 3.0) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-106:
		tmp = 2.0 / ((t_m * k) * (math.pow(k, 3.0) / math.pow(l, 2.0)))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((t_m * (math.pow(t_m, 2.0) / l)) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-106)
		tmp = Float64(2.0 / Float64(Float64(t_m * k) * Float64((k ^ 3.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-106)
		tmp = 2.0 / ((t_m * k) * ((k ^ 3.0) / (l ^ 2.0)));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * ((t_m * ((t_m ^ 2.0) / l)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-106], N[(2.0 / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3000000000000001e-106

   1. Initial program 41.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 41.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.2%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.2%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.5%

       \[\leadsto \frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot t\right)}} \]

   if 2.3000000000000001e-106 < t

   1. Initial program 65.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 66.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 61.2%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. cube-mult 61.2%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      2. *-un-lft-identity 61.2%

         \[\leadsto \frac{2}{\frac{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      3. times-frac 62.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{t \cdot t}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

      4. pow2 62.5%

         \[\leadsto \frac{2}{\frac{\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]

   7. Applied egg-rr 62.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.2e-113)
    (/ 2.0 (* (* t_m k) (/ (pow k 3.0) (pow l 2.0))))
    (/ 2.0 (/ (* (* 2.0 (pow k 2.0)) (/ (pow t_m 3.0) l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-113) {
		tmp = 2.0 / ((t_m * k) * (pow(k, 3.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.2d-113) then
        tmp = 2.0d0 / ((t_m * k) * ((k ** 3.0d0) / (l ** 2.0d0)))
    else
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-113) {
		tmp = 2.0 / ((t_m * k) * (Math.pow(k, 3.0) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.2e-113:
		tmp = 2.0 / ((t_m * k) * (math.pow(k, 3.0) / math.pow(l, 2.0)))
	else:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.2e-113)
		tmp = Float64(2.0 / Float64(Float64(t_m * k) * Float64((k ^ 3.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.2e-113)
		tmp = 2.0 / ((t_m * k) * ((k ^ 3.0) / (l ^ 2.0)));
	else
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-113], N[(2.0 / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.20000000000000004e-113

   1. Initial program 42.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.9%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.9%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.8%

       \[\leadsto \frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot t\right)}} \]

   if 2.20000000000000004e-113 < t

   1. Initial program 62.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.1%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.2e-113)
    (/ 2.0 (* (* t_m k) (/ (pow k 3.0) (pow l 2.0))))
    (/ 2.0 (* (/ (* 2.0 (pow k 2.0)) l) (/ (pow t_m 3.0) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-113) {
		tmp = 2.0 / ((t_m * k) * (pow(k, 3.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) / l) * (pow(t_m, 3.0) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.2d-113) then
        tmp = 2.0d0 / ((t_m * k) * ((k ** 3.0d0) / (l ** 2.0d0)))
    else
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) / l) * ((t_m ** 3.0d0) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-113) {
		tmp = 2.0 / ((t_m * k) * (Math.pow(k, 3.0) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) / l) * (Math.pow(t_m, 3.0) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.2e-113:
		tmp = 2.0 / ((t_m * k) * (math.pow(k, 3.0) / math.pow(l, 2.0)))
	else:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) / l) * (math.pow(t_m, 3.0) / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.2e-113)
		tmp = Float64(2.0 / Float64(Float64(t_m * k) * Float64((k ^ 3.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64((t_m ^ 3.0) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.2e-113)
		tmp = 2.0 / ((t_m * k) * ((k ^ 3.0) / (l ^ 2.0)));
	else
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) / l) * ((t_m ^ 3.0) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-113], N[(2.0 / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.20000000000000004e-113

   1. Initial program 42.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.9%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.9%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.8%

       \[\leadsto \frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot t\right)}} \]

   if 2.20000000000000004e-113 < t

   1. Initial program 62.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.1%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 59.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{{t\_m}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.2e-113)
    (/ 2.0 (* (* t_m k) (/ (pow k 3.0) (pow l 2.0))))
    (* l (/ 2.0 (* (pow t_m 3.0) (/ (* 2.0 (pow k 2.0)) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-113) {
		tmp = 2.0 / ((t_m * k) * (pow(k, 3.0) / pow(l, 2.0)));
	} else {
		tmp = l * (2.0 / (pow(t_m, 3.0) * ((2.0 * pow(k, 2.0)) / l)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.2d-113) then
        tmp = 2.0d0 / ((t_m * k) * ((k ** 3.0d0) / (l ** 2.0d0)))
    else
        tmp = l * (2.0d0 / ((t_m ** 3.0d0) * ((2.0d0 * (k ** 2.0d0)) / l)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-113) {
		tmp = 2.0 / ((t_m * k) * (Math.pow(k, 3.0) / Math.pow(l, 2.0)));
	} else {
		tmp = l * (2.0 / (Math.pow(t_m, 3.0) * ((2.0 * Math.pow(k, 2.0)) / l)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.2e-113:
		tmp = 2.0 / ((t_m * k) * (math.pow(k, 3.0) / math.pow(l, 2.0)))
	else:
		tmp = l * (2.0 / (math.pow(t_m, 3.0) * ((2.0 * math.pow(k, 2.0)) / l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.2e-113)
		tmp = Float64(2.0 / Float64(Float64(t_m * k) * Float64((k ^ 3.0) / (l ^ 2.0))));
	else
		tmp = Float64(l * Float64(2.0 / Float64((t_m ^ 3.0) * Float64(Float64(2.0 * (k ^ 2.0)) / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.2e-113)
		tmp = 2.0 / ((t_m * k) * ((k ^ 3.0) / (l ^ 2.0)));
	else
		tmp = l * (2.0 / ((t_m ^ 3.0) * ((2.0 * (k ^ 2.0)) / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-113], N[(2.0 / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.20000000000000004e-113

   1. Initial program 42.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 40.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-/l* 40.9%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   7. Simplified 40.9%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   8. Taylor expanded in k around inf 52.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
   9. Step-by-step derivation

      1. times-frac 52.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   10. Simplified 52.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

   11. Taylor expanded in k around 0 52.8%

       \[\leadsto \frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot t\right)}} \]

   if 2.20000000000000004e-113 < t

   1. Initial program 62.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 59.1%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 59.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 59.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 59.8%

         \[\leadsto \color{blue}{1 \cdot \frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

      2. associate-/r/ 59.8%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \ell\right)} \]

   9. Applied egg-rr 59.8%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-lft-identity 59.8%

          \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)} \cdot \ell} \]

       2. *-commutative 59.8%

          \[\leadsto \color{blue}{\ell \cdot \frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}} \]

       3. associate-*l/ 59.0%

          \[\leadsto \ell \cdot \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

       4. associate-/l* 59.6%

          \[\leadsto \ell \cdot \frac{2}{\color{blue}{{t}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   11. Simplified 59.6%

       \[\leadsto \color{blue}{\ell \cdot \frac{2}{{t}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(t \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{{t}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(t\_m \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* t_m k) (/ (pow k 3.0) (pow l 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((t_m * k) * (pow(k, 3.0) / pow(l, 2.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((t_m * k) * ((k ** 3.0d0) / (l ** 2.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((t_m * k) * (Math.pow(k, 3.0) / Math.pow(l, 2.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((t_m * k) * (math.pow(k, 3.0) / math.pow(l, 2.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(t_m * k) * Float64((k ^ 3.0) / (l ^ 2.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((t_m * k) * ((k ^ 3.0) / (l ^ 2.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 48.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 47.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-/l* 47.3%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

7. Simplified 47.3%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

8. Taylor expanded in k around inf 51.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
9. Step-by-step derivation

   1. times-frac 51.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

10. Simplified 51.4%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

11. Taylor expanded in k around 0 50.9%

    \[\leadsto \frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \color{blue}{\left(k \cdot t\right)}} \]

12. Final simplification 50.9%

    \[\leadsto \frac{2}{\left(t \cdot k\right) \cdot \frac{{k}^{3}}{{\ell}^{2}}} \]
13. Add Preprocessing

Alternative 27: 51.1% 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 #s(literal 1 binary64) 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 48.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 48.8%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 47.1%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-/l* 47.3%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

7. Simplified 47.3%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

8. Taylor expanded in k around inf 51.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2} \cdot \cos k}}} \]
9. Step-by-step derivation

   1. times-frac 51.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

10. Simplified 51.4%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{t \cdot \sin k}{\cos k}}} \]

11. Taylor expanded in k around 0 50.1%

    \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

12. Final simplification 50.1%

    \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))