Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.3% → 79.7%

Time: 26.9s

Alternatives: 22

Speedup: 2.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% 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: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left(1 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_3 := \frac{t_m}{\sqrt[3]{\ell}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.38 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t_m \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({t_3}^{2} \cdot \left(t_3 \cdot \frac{1}{\ell}\right)\right)\right) \cdot \left(\tan k \cdot t_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))) (t_3 (/ t_m (cbrt l))))
   (*
    t_s
    (if (<= t_m 1.38e-70)
      (*
       2.0
       (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
      (if (<= t_m 3.3e+96)
        (/ 2.0 (* (* (/ (/ (pow t_m 3.0) l) (/ l (sin k))) (tan k)) t_2))
        (/
         2.0
         (*
          (* (sin k) (* (pow t_3 2.0) (* t_3 (/ 1.0 l))))
          (* (tan k) t_2))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (1.0 + pow((k / t_m), 2.0));
	double t_3 = t_m / cbrt(l);
	double tmp;
	if (t_m <= 1.38e-70) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l) / (l / sin(k))) * tan(k)) * t_2);
	} else {
		tmp = 2.0 / ((sin(k) * (pow(t_3, 2.0) * (t_3 * (1.0 / l)))) * (tan(k) * t_2));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (1.0 + Math.pow((k / t_m), 2.0));
	double t_3 = t_m / Math.cbrt(l);
	double tmp;
	if (t_m <= 1.38e-70) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 3.3e+96) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l) / (l / Math.sin(k))) * Math.tan(k)) * t_2);
	} else {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_3, 2.0) * (t_3 * (1.0 / l)))) * (Math.tan(k) * t_2));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))
	t_3 = Float64(t_m / cbrt(l))
	tmp = 0.0
	if (t_m <= 1.38e-70)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 3.3e+96)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l) / Float64(l / sin(k))) * tan(k)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_3 ^ 2.0) * Float64(t_3 * Float64(1.0 / l)))) * Float64(tan(k) * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.38e-70], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.3e+96], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$3, 2.0], $MachinePrecision] * N[(t$95$3 * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3800000000000001e-70

   1. Initial program 49.2%

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

      1. associate-/r* 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-/r* 51.6%

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

      6. associate-+l+ 51.6%

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

      7. unpow2 51.6%

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

      8. times-frac 38.6%

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

      9. sqr-neg 38.6%

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

      10. times-frac 51.6%

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

      11. unpow2 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*r* 68.9%

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

      3. times-frac 69.8%

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

   7. Simplified 69.8%

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

   if 1.3800000000000001e-70 < t < 3.29999999999999984e96

   1. Initial program 67.2%

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

      1. associate-/r* 89.6%

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

      2. associate-*l/ 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. associate-/l* 92.9%

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

   6. Simplified 92.9%

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

   if 3.29999999999999984e96 < t

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

      2. sqr-neg 65.1%

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

      3. *-commutative 65.1%

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

      4. associate-*l* 65.1%

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

      5. *-commutative 65.1%

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

      6. sqr-neg 65.1%

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

   3. Simplified 65.1%

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

      1. associate-/r* 71.2%

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

      2. div-inv 71.2%

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

      3. add-cube-cbrt 71.2%

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

      4. associate-*l* 71.2%

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

      5. pow2 71.2%

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

      6. cbrt-div 71.2%

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

      7. rem-cbrt-cube 71.2%

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

      8. cbrt-div 71.2%

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

      9. rem-cbrt-cube 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.38 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left({\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{2} \cdot \left(\frac{t}{\sqrt[3]{\ell}} \cdot \frac{1}{\ell}\right)\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 + \left(1 + t_2\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{\ell \cdot \ell}\right)\right) \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right) \cdot \left(\tan k \cdot \left(2 + t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (+ 1.0 (+ 1.0 t_2))
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))
         5e+263)
      (/ 2.0 (* (* (sin k) (/ (/ (pow t_m 3.0) l) l)) (* (tan k) (+ 2.0 t_2))))
      (/
       2.0
       (pow
        (*
         (/ (pow t_m 1.5) l)
         (* (sqrt 2.0) (+ k (* (pow k 3.0) -0.08333333333333333))))
        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 (((1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))))) <= 5e+263) {
		tmp = 2.0 / ((sin(k) * ((pow(t_m, 3.0) / l) / l)) * (tan(k) * (2.0 + t_2)));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (sqrt(2.0) * (k + (pow(k, 3.0) * -0.08333333333333333)))), 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 (((1.0d0 + (1.0d0 + t_2)) * (tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))) <= 5d+263) then
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 3.0d0) / l) / l)) * (tan(k) * (2.0d0 + t_2)))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (sqrt(2.0d0) * (k + ((k ** 3.0d0) * (-0.08333333333333333d0))))) ** 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 (((1.0 + (1.0 + t_2)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))))) <= 5e+263) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l)) * (Math.tan(k) * (2.0 + t_2)));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.sqrt(2.0) * (k + (Math.pow(k, 3.0) * -0.08333333333333333)))), 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 ((1.0 + (1.0 + t_2)) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))) <= 5e+263:
		tmp = 2.0 / ((math.sin(k) * ((math.pow(t_m, 3.0) / l) / l)) * (math.tan(k) * (2.0 + t_2)))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (math.sqrt(2.0) * (k + (math.pow(k, 3.0) * -0.08333333333333333)))), 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 (Float64(Float64(1.0 + Float64(1.0 + t_2)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))))) <= 5e+263)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l)) * Float64(tan(k) * Float64(2.0 + t_2))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(sqrt(2.0) * Float64(k + Float64((k ^ 3.0) * -0.08333333333333333)))) ^ 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 (((1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l))))) <= 5e+263)
		tmp = 2.0 / ((sin(k) * (((t_m ^ 3.0) / l) / l)) * (tan(k) * (2.0 + t_2)));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (sqrt(2.0) * (k + ((k ^ 3.0) * -0.08333333333333333)))) ^ 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[N[(N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+263], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(k + N[(N[Power[k, 3.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 5.00000000000000022e263

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. sqr-neg 83.7%

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

      3. sqr-neg 83.7%

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

      4. associate-/r* 91.5%

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

      5. distribute-rgt-in 91.5%

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

      6. unpow2 91.5%

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

      7. times-frac 74.1%

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

      8. sqr-neg 74.1%

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

      9. times-frac 91.5%

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

      10. unpow2 91.5%

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

      11. distribute-rgt-in 91.5%

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

   3. Simplified 91.5%

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

   if 5.00000000000000022e263 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 26.2%

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

      1. *-commutative 26.2%

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

      2. sqr-neg 26.2%

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

      3. *-commutative 26.2%

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

      4. associate-*l* 26.2%

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

      5. *-commutative 26.2%

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

      6. sqr-neg 26.2%

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

   3. Simplified 26.2%

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

   5. Taylor expanded in k around 0 32.0%

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

      1. add-sqr-sqrt 24.6%

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

      2. pow2 24.6%

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

   7. Applied egg-rr 16.7%

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

      1. *-commutative 16.7%

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

   9. Simplified 16.7%

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

   10. Taylor expanded in k around 0 28.6%

       \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left({k}^{3} \cdot \sqrt{2}\right) + k \cdot \sqrt{2}\right)}\right)}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 28.6%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k \cdot \sqrt{2} + -0.08333333333333333 \cdot \left({k}^{3} \cdot \sqrt{2}\right)\right)}\right)}^{2}} \]

       2. associate-*r* 28.6%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2} + \color{blue}{\left(-0.08333333333333333 \cdot {k}^{3}\right) \cdot \sqrt{2}}\right)\right)}^{2}} \]

       3. distribute-rgt-out 28.6%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(k + -0.08333333333333333 \cdot {k}^{3}\right)\right)}\right)}^{2}} \]

       4. *-commutative 28.6%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + \color{blue}{{k}^{3} \cdot -0.08333333333333333}\right)\right)\right)}^{2}} \]

   12. Simplified 28.6%

       \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left(1 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.08 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t_m \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t_m}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 1.08e-70)
      (*
       2.0
       (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
      (if (<= t_m 5e+86)
        (/ 2.0 (* (* (/ (/ (pow t_m 3.0) l) (/ l (sin k))) (tan k)) t_2))
        (/
         2.0
         (* t_2 (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0))))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (1.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 1.08e-70) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 5e+86) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l) / (l / sin(k))) * tan(k)) * t_2);
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))
    if (t_m <= 1.08d-70) then
        tmp = 2.0d0 * ((cos(k) / (t_m * (k ** 2.0d0))) * ((l ** 2.0d0) / (sin(k) ** 2.0d0)))
    else if (t_m <= 5d+86) then
        tmp = 2.0d0 / (((((t_m ** 3.0d0) / l) / (l / sin(k))) * tan(k)) * t_2)
    else
        tmp = 2.0d0 / (t_2 * (tan(k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (1.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 1.08e-70) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 5e+86) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l) / (l / Math.sin(k))) * Math.tan(k)) * t_2);
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 1.0 + (1.0 + math.pow((k / t_m), 2.0))
	tmp = 0
	if t_m <= 1.08e-70:
		tmp = 2.0 * ((math.cos(k) / (t_m * math.pow(k, 2.0))) * (math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)))
	elif t_m <= 5e+86:
		tmp = 2.0 / ((((math.pow(t_m, 3.0) / l) / (l / math.sin(k))) * math.tan(k)) * t_2)
	else:
		tmp = 2.0 / (t_2 * (math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 1.08e-70)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 5e+86)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l) / Float64(l / sin(k))) * tan(k)) * t_2));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 1.0 + (1.0 + ((k / t_m) ^ 2.0));
	tmp = 0.0;
	if (t_m <= 1.08e-70)
		tmp = 2.0 * ((cos(k) / (t_m * (k ^ 2.0))) * ((l ^ 2.0) / (sin(k) ^ 2.0)));
	elseif (t_m <= 5e+86)
		tmp = 2.0 / (((((t_m ^ 3.0) / l) / (l / sin(k))) * tan(k)) * t_2);
	else
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.08e-70], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+86], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.0800000000000001e-70

   1. Initial program 49.2%

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

      1. associate-/r* 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-/r* 51.6%

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

      6. associate-+l+ 51.6%

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

      7. unpow2 51.6%

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

      8. times-frac 38.6%

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

      9. sqr-neg 38.6%

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

      10. times-frac 51.6%

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

      11. unpow2 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*r* 68.9%

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

      3. times-frac 69.8%

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

   7. Simplified 69.8%

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

   if 1.0800000000000001e-70 < t < 4.9999999999999998e86

   1. Initial program 74.3%

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

      1. associate-/r* 92.3%

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

      2. associate-*l/ 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. associate-/l* 96.0%

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

   6. Simplified 96.0%

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

   if 4.9999999999999998e86 < t

   1. Initial program 60.3%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. sqrt-div 60.3%

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

      4. sqrt-pow1 60.9%

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

      5. metadata-eval 60.9%

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

      6. sqrt-prod 38.0%

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

      7. add-sqr-sqrt 86.4%

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

   4. Applied egg-rr 86.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t_m}^{1.5}}{\ell}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.1 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t_m \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(t_2 \cdot t_2\right)\right) \cdot \frac{2 \cdot \sin k}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 4.1e-70)
      (*
       2.0
       (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
      (if (<= t_m 5.5e+86)
        (/
         2.0
         (*
          (* (/ (/ (pow t_m 3.0) l) (/ l (sin k))) (tan k))
          (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
        (/ 2.0 (* (* (sin k) (* t_2 t_2)) (/ (* 2.0 (sin k)) (cos k)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 4.1e-70) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 5.5e+86) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l) / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / ((sin(k) * (t_2 * t_2)) * ((2.0 * sin(k)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m ** 1.5d0) / l
    if (t_m <= 4.1d-70) then
        tmp = 2.0d0 * ((cos(k) / (t_m * (k ** 2.0d0))) * ((l ** 2.0d0) / (sin(k) ** 2.0d0)))
    else if (t_m <= 5.5d+86) then
        tmp = 2.0d0 / (((((t_m ** 3.0d0) / l) / (l / sin(k))) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    else
        tmp = 2.0d0 / ((sin(k) * (t_2 * t_2)) * ((2.0d0 * sin(k)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 4.1e-70) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 5.5e+86) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l) / (l / Math.sin(k))) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * (t_2 * t_2)) * ((2.0 * Math.sin(k)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(t_m, 1.5) / l
	tmp = 0
	if t_m <= 4.1e-70:
		tmp = 2.0 * ((math.cos(k) / (t_m * math.pow(k, 2.0))) * (math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)))
	elif t_m <= 5.5e+86:
		tmp = 2.0 / ((((math.pow(t_m, 3.0) / l) / (l / math.sin(k))) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	else:
		tmp = 2.0 / ((math.sin(k) * (t_2 * t_2)) * ((2.0 * math.sin(k)) / math.cos(k)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 4.1e-70)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 5.5e+86)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l) / Float64(l / sin(k))) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(t_2 * t_2)) * Float64(Float64(2.0 * sin(k)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (t_m <= 4.1e-70)
		tmp = 2.0 * ((cos(k) / (t_m * (k ^ 2.0))) * ((l ^ 2.0) / (sin(k) ^ 2.0)));
	elseif (t_m <= 5.5e+86)
		tmp = 2.0 / (((((t_m ^ 3.0) / l) / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	else
		tmp = 2.0 / ((sin(k) * (t_2 * t_2)) * ((2.0 * sin(k)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.1e-70], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+86], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.09999999999999977e-70

   1. Initial program 49.2%

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

      1. associate-/r* 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-/r* 51.6%

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

      6. associate-+l+ 51.6%

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

      7. unpow2 51.6%

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

      8. times-frac 38.6%

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

      9. sqr-neg 38.6%

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

      10. times-frac 51.6%

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

      11. unpow2 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*r* 68.9%

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

      3. times-frac 69.8%

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

   7. Simplified 69.8%

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

   if 4.09999999999999977e-70 < t < 5.5000000000000002e86

   1. Initial program 74.3%

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

      1. associate-/r* 92.3%

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

      2. associate-*l/ 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. associate-/l* 96.0%

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

   6. Simplified 96.0%

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

   if 5.5000000000000002e86 < t

   1. Initial program 60.3%

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

      1. associate-*l* 60.3%

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

      2. sqr-neg 60.3%

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

      3. sqr-neg 60.3%

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

      4. associate-/r* 70.8%

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

      5. distribute-rgt-in 70.8%

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

      6. unpow2 70.8%

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

      7. times-frac 62.7%

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

      8. sqr-neg 62.7%

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

      9. times-frac 70.8%

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

      10. unpow2 70.8%

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

      11. distribute-rgt-in 70.8%

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

   3. Simplified 70.8%

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

      1. cube-mult 70.9%

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

      2. *-un-lft-identity 70.9%

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

      3. times-frac 73.5%

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

      4. pow2 73.5%

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

   6. Applied egg-rr 73.5%

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

   7. Taylor expanded in t around inf 73.5%

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

      1. associate-*r/ 73.5%

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

   9. Simplified 73.5%

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

       1. /-rgt-identity 73.5%

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

       2. associate-*r/ 70.8%

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

       3. unpow2 70.8%

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

       4. cube-mult 70.8%

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

       5. associate-/r* 60.3%

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

       6. add-sqr-sqrt 60.3%

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

       7. sqrt-div 60.3%

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

       8. sqrt-pow1 60.3%

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

       9. metadata-eval 60.3%

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

       10. sqrt-prod 24.5%

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

       11. add-sqr-sqrt 59.8%

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

       12. sqrt-div 59.8%

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

       13. sqrt-pow1 60.2%

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

       14. metadata-eval 60.2%

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

       15. sqrt-prod 35.7%

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

       16. add-sqr-sqrt 83.9%

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

   11. Applied egg-rr 83.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)\right) \cdot \frac{2 \cdot \sin k}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.4 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t_m \leq 3.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \sin k}{\cos k} \cdot \left(\sin k \cdot \frac{{\left(\frac{t_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.4e-70)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (if (<= t_m 3.8e+94)
      (/
       2.0
       (*
        (* (/ (/ (pow t_m 3.0) l) (/ l (sin k))) (tan k))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (/
       2.0
       (*
        (/ (* 2.0 (sin k)) (cos k))
        (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-70) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 3.8e+94) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l) / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / (((2.0 * sin(k)) / cos(k)) * (sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.4e-70) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 3.8e+94) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l) / (l / Math.sin(k))) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / (((2.0 * Math.sin(k)) / Math.cos(k)) * (Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.4e-70)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 3.8e+94)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l) / Float64(l / sin(k))) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * sin(k)) / cos(k)) * Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.4e-70], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.8e+94], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.39999999999999995e-70

   1. Initial program 49.2%

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

      1. associate-/r* 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-/r* 51.6%

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

      6. associate-+l+ 51.6%

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

      7. unpow2 51.6%

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

      8. times-frac 38.6%

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

      9. sqr-neg 38.6%

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

      10. times-frac 51.6%

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

      11. unpow2 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*r* 68.9%

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

      3. times-frac 69.8%

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

   7. Simplified 69.8%

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

   if 3.39999999999999995e-70 < t < 3.7999999999999996e94

   1. Initial program 69.4%

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

      1. associate-/r* 89.3%

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

      2. associate-*l/ 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. associate-/l* 92.7%

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

   6. Simplified 92.7%

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

   if 3.7999999999999996e94 < t

   1. Initial program 63.4%

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

      1. associate-*l* 63.4%

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

      2. sqr-neg 63.4%

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

      3. sqr-neg 63.4%

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

      4. associate-/r* 72.0%

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

      5. distribute-rgt-in 72.0%

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

      6. unpow2 72.0%

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

      7. times-frac 63.4%

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

      8. sqr-neg 63.4%

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

      9. times-frac 72.0%

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

      10. unpow2 72.0%

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

      11. distribute-rgt-in 72.0%

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

   3. Simplified 72.0%

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

      1. add-cube-cbrt 72.0%

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

      2. pow3 72.0%

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

      3. cbrt-div 72.0%

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

      4. rem-cbrt-cube 85.6%

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

   6. Applied egg-rr 85.6%

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

   7. Taylor expanded in t around inf 83.1%

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

      1. associate-*r/ 74.8%

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

   9. Simplified 83.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \sin k}{\cos k} \cdot \left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.08 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_m \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t_m \leq 7.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.08e-70)
    (*
     2.0
     (* (/ (cos k) (* t_m (pow k 2.0))) (/ (pow l 2.0) (pow (sin k) 2.0))))
    (if (<= t_m 7.4e+124)
      (/
       2.0
       (*
        (* (/ (/ (pow t_m 3.0) l) (/ l (sin k))) (tan k))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
      (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 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 <= 1.08e-70) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(k, 2.0))) * (pow(l, 2.0) / pow(sin(k), 2.0)));
	} else if (t_m <= 7.4e+124) {
		tmp = 2.0 / ((((pow(t_m, 3.0) / l) / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 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 <= 1.08d-70) then
        tmp = 2.0d0 * ((cos(k) / (t_m * (k ** 2.0d0))) * ((l ** 2.0d0) / (sin(k) ** 2.0d0)))
    else if (t_m <= 7.4d+124) then
        tmp = 2.0d0 / (((((t_m ** 3.0d0) / l) / (l / sin(k))) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 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 <= 1.08e-70) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(k, 2.0))) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k), 2.0)));
	} else if (t_m <= 7.4e+124) {
		tmp = 2.0 / ((((Math.pow(t_m, 3.0) / l) / (l / Math.sin(k))) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 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 <= 1.08e-70:
		tmp = 2.0 * ((math.cos(k) / (t_m * math.pow(k, 2.0))) * (math.pow(l, 2.0) / math.pow(math.sin(k), 2.0)))
	elif t_m <= 7.4e+124:
		tmp = 2.0 / ((((math.pow(t_m, 3.0) / l) / (l / math.sin(k))) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 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 <= 1.08e-70)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (k ^ 2.0))) * Float64((l ^ 2.0) / (sin(k) ^ 2.0))));
	elseif (t_m <= 7.4e+124)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / l) / Float64(l / sin(k))) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 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 <= 1.08e-70)
		tmp = 2.0 * ((cos(k) / (t_m * (k ^ 2.0))) * ((l ^ 2.0) / (sin(k) ^ 2.0)));
	elseif (t_m <= 7.4e+124)
		tmp = 2.0 / (((((t_m ^ 3.0) / l) / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 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, 1.08e-70], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+124], N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.0800000000000001e-70

   1. Initial program 49.2%

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

      1. associate-/r* 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 45.9%

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

      4. sqr-neg 45.9%

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

      5. associate-/r* 51.6%

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

      6. associate-+l+ 51.6%

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

      7. unpow2 51.6%

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

      8. times-frac 38.6%

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

      9. sqr-neg 38.6%

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

      10. times-frac 51.6%

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

      11. unpow2 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. associate-*r* 68.9%

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

      3. times-frac 69.8%

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

   7. Simplified 69.8%

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

   if 1.0800000000000001e-70 < t < 7.40000000000000016e124

   1. Initial program 68.2%

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

      1. associate-/r* 87.9%

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

      2. associate-*l/ 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. associate-/l* 90.8%

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

   6. Simplified 90.8%

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

   if 7.40000000000000016e124 < t

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. sqr-neg 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. *-commutative 63.7%

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

      6. sqr-neg 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in k around 0 63.7%

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

      1. add-sqr-sqrt 50.4%

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

      2. pow2 50.4%

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

   7. Applied egg-rr 43.0%

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

      1. *-commutative 43.0%

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

   9. Simplified 43.0%

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

   10. Taylor expanded in k around 0 74.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{-70}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t_m}^{3}}{\ell}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{t_2}{\ell}\right)}{\cos k}}\\ \mathbf{elif}\;t_m \leq 1.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{\sin k}{\frac{\cos k}{t_m}}}\\ \mathbf{elif}\;t_m \leq 7.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\left(\frac{t_2}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 3.0) l)))
   (*
    t_s
    (if (<= t_m 3.5e-216)
      (/ 2.0 (/ (* (* 2.0 (sin k)) (* (sin k) (/ t_2 l))) (cos k)))
      (if (<= t_m 1.6e-87)
        (/ 2.0 (* (/ (pow k 3.0) (pow l 2.0)) (/ (sin k) (/ (cos k) t_m))))
        (if (<= t_m 7.4e+124)
          (/
           2.0
           (*
            (* (/ t_2 (/ l (sin k))) (tan k))
            (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
          (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 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(t_m, 3.0) / l;
	double tmp;
	if (t_m <= 3.5e-216) {
		tmp = 2.0 / (((2.0 * sin(k)) * (sin(k) * (t_2 / l))) / cos(k));
	} else if (t_m <= 1.6e-87) {
		tmp = 2.0 / ((pow(k, 3.0) / pow(l, 2.0)) * (sin(k) / (cos(k) / t_m)));
	} else if (t_m <= 7.4e+124) {
		tmp = 2.0 / (((t_2 / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 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 = (t_m ** 3.0d0) / l
    if (t_m <= 3.5d-216) then
        tmp = 2.0d0 / (((2.0d0 * sin(k)) * (sin(k) * (t_2 / l))) / cos(k))
    else if (t_m <= 1.6d-87) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l ** 2.0d0)) * (sin(k) / (cos(k) / t_m)))
    else if (t_m <= 7.4d+124) then
        tmp = 2.0d0 / (((t_2 / (l / sin(k))) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 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(t_m, 3.0) / l;
	double tmp;
	if (t_m <= 3.5e-216) {
		tmp = 2.0 / (((2.0 * Math.sin(k)) * (Math.sin(k) * (t_2 / l))) / Math.cos(k));
	} else if (t_m <= 1.6e-87) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / Math.pow(l, 2.0)) * (Math.sin(k) / (Math.cos(k) / t_m)));
	} else if (t_m <= 7.4e+124) {
		tmp = 2.0 / (((t_2 / (l / Math.sin(k))) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 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(t_m, 3.0) / l
	tmp = 0
	if t_m <= 3.5e-216:
		tmp = 2.0 / (((2.0 * math.sin(k)) * (math.sin(k) * (t_2 / l))) / math.cos(k))
	elif t_m <= 1.6e-87:
		tmp = 2.0 / ((math.pow(k, 3.0) / math.pow(l, 2.0)) * (math.sin(k) / (math.cos(k) / t_m)))
	elif t_m <= 7.4e+124:
		tmp = 2.0 / (((t_2 / (l / math.sin(k))) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 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((t_m ^ 3.0) / l)
	tmp = 0.0
	if (t_m <= 3.5e-216)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * sin(k)) * Float64(sin(k) * Float64(t_2 / l))) / cos(k)));
	elseif (t_m <= 1.6e-87)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / (l ^ 2.0)) * Float64(sin(k) / Float64(cos(k) / t_m))));
	elseif (t_m <= 7.4e+124)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 / Float64(l / sin(k))) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 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 = (t_m ^ 3.0) / l;
	tmp = 0.0;
	if (t_m <= 3.5e-216)
		tmp = 2.0 / (((2.0 * sin(k)) * (sin(k) * (t_2 / l))) / cos(k));
	elseif (t_m <= 1.6e-87)
		tmp = 2.0 / (((k ^ 3.0) / (l ^ 2.0)) * (sin(k) / (cos(k) / t_m)));
	elseif (t_m <= 7.4e+124)
		tmp = 2.0 / (((t_2 / (l / sin(k))) * tan(k)) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 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[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.5e-216], N[(2.0 / N[(N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e-87], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+124], N[(2.0 / N[(N[(N[(t$95$2 / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 3.49999999999999982e-216

   1. Initial program 49.0%

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

      1. associate-*l* 49.0%

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

      2. sqr-neg 49.0%

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

      3. sqr-neg 49.0%

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

      4. associate-/r* 55.3%

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

      5. distribute-rgt-in 55.3%

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

      6. unpow2 55.3%

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

      7. times-frac 42.2%

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

      8. sqr-neg 42.2%

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

      9. times-frac 55.3%

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

      10. unpow2 55.3%

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

      11. distribute-rgt-in 55.3%

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

   3. Simplified 55.3%

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

      1. cube-mult 55.3%

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

      2. *-un-lft-identity 55.3%

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

      3. times-frac 61.6%

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

      4. pow2 61.6%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in t around inf 56.3%

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

      1. associate-*r/ 56.3%

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

   9. Simplified 56.3%

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

       1. associate-*r/ 56.3%

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

       2. frac-times 54.9%

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

       3. unpow2 54.9%

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

       4. cube-mult 54.9%

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

       5. *-un-lft-identity 54.9%

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

   11. Applied egg-rr 54.9%

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

   if 3.49999999999999982e-216 < t < 1.59999999999999989e-87

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

      2. sqr-neg 49.9%

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

      3. *-commutative 49.9%

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

      4. associate-*l* 49.9%

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

      5. *-commutative 49.9%

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

      6. sqr-neg 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in k around 0 46.5%

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

   6. Taylor expanded in k around inf 68.5%

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

      1. times-frac 68.6%

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

      2. *-commutative 68.6%

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

      3. associate-/l* 68.6%

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

   8. Simplified 68.6%

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

   if 1.59999999999999989e-87 < t < 7.40000000000000016e124

   1. Initial program 66.4%

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

      1. associate-/r* 84.0%

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

      2. associate-*l/ 89.2%

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

   4. Applied egg-rr 89.2%

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

      1. associate-/l* 89.2%

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

   6. Simplified 89.2%

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

   if 7.40000000000000016e124 < t

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. sqr-neg 63.7%

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

      3. *-commutative 63.7%

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

      4. associate-*l* 63.7%

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

      5. *-commutative 63.7%

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

      6. sqr-neg 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in k around 0 63.7%

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

      1. add-sqr-sqrt 50.4%

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

      2. pow2 50.4%

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

   7. Applied egg-rr 43.0%

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

      1. *-commutative 43.0%

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

   9. Simplified 43.0%

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

   10. Taylor expanded in k around 0 74.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{\sin k}{\frac{\cos k}{t}}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}} \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t_m \cdot \sqrt[3]{{k}^{2} \cdot {\ell}^{-2}}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.5e-58)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (if (<= k 4.9e+147)
      (/
       2.0
       (/ (* (* 2.0 (sin k)) (* (sin k) (/ (/ (pow t_m 3.0) l) l))) (cos k)))
      (pow (/ 1.0 (* t_m (cbrt (* (pow k 2.0) (pow 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 tmp;
	if (k <= 8.5e-58) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else if (k <= 4.9e+147) {
		tmp = 2.0 / (((2.0 * sin(k)) * (sin(k) * ((pow(t_m, 3.0) / l) / l))) / cos(k));
	} else {
		tmp = pow((1.0 / (t_m * cbrt((pow(k, 2.0) * pow(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 tmp;
	if (k <= 8.5e-58) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else if (k <= 4.9e+147) {
		tmp = 2.0 / (((2.0 * Math.sin(k)) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l))) / Math.cos(k));
	} else {
		tmp = Math.pow((1.0 / (t_m * Math.cbrt((Math.pow(k, 2.0) * Math.pow(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)
	tmp = 0.0
	if (k <= 8.5e-58)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	elseif (k <= 4.9e+147)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * sin(k)) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))) / cos(k)));
	else
		tmp = Float64(1.0 / Float64(t_m * cbrt(Float64((k ^ 2.0) * (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_] := N[(t$95$s * If[LessEqual[k, 8.5e-58], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.9e+147], N[(2.0 / N[(N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(t$95$m * N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 8.5000000000000004e-58

   1. Initial program 54.5%

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

      1. *-commutative 54.5%

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

      2. sqr-neg 54.5%

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

      3. *-commutative 54.5%

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

      4. associate-*l* 54.5%

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

      5. *-commutative 54.5%

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

      6. sqr-neg 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in k around 0 54.2%

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

      1. add-sqr-sqrt 26.7%

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

      2. pow2 26.7%

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

   7. Applied egg-rr 23.0%

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

      1. *-commutative 23.0%

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

   9. Simplified 23.0%

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

   10. Taylor expanded in k around 0 29.1%

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

   if 8.5000000000000004e-58 < k < 4.8999999999999998e147

   1. Initial program 56.6%

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

      1. associate-*l* 56.7%

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

      2. sqr-neg 56.7%

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

      3. sqr-neg 56.7%

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

      4. associate-/r* 66.3%

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

      5. distribute-rgt-in 66.3%

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

      6. unpow2 66.3%

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

      7. times-frac 66.3%

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

      8. sqr-neg 66.3%

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

      9. times-frac 66.3%

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

      10. unpow2 66.3%

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

      11. distribute-rgt-in 66.3%

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

   3. Simplified 66.3%

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

      1. cube-mult 66.3%

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

      2. *-un-lft-identity 66.3%

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

      3. times-frac 73.5%

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

      4. pow2 73.5%

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

   6. Applied egg-rr 73.5%

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

   7. Taylor expanded in t around inf 62.6%

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

      1. associate-*r/ 62.6%

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

   9. Simplified 62.6%

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

       1. associate-*r/ 62.6%

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

       2. frac-times 60.0%

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

       3. unpow2 60.0%

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

       4. cube-mult 60.0%

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

       5. *-un-lft-identity 60.0%

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

   11. Applied egg-rr 60.0%

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

   if 4.8999999999999998e147 < k

   1. Initial program 45.3%

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

      1. *-commutative 45.3%

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

      2. sqr-neg 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-*l* 45.3%

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

      5. *-commutative 45.3%

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

      6. sqr-neg 45.3%

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

   3. Simplified 45.3%

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

      1. associate-/r* 45.3%

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

      2. div-inv 45.3%

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

      3. add-cube-cbrt 45.3%

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

      4. associate-*l* 45.3%

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

      5. pow2 45.3%

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

      6. cbrt-div 45.3%

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

      7. rem-cbrt-cube 45.3%

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

      8. cbrt-div 45.3%

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

      9. rem-cbrt-cube 67.0%

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

   6. Applied egg-rr 67.0%

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

   7. Taylor expanded in k around 0 45.2%

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

      1. associate-/l* 45.2%

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

      2. associate-/r/ 45.2%

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

   9. Simplified 45.2%

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

       1. add-cube-cbrt 45.2%

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

       2. pow3 45.2%

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

   11. Applied egg-rr 69.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t \cdot \sqrt[3]{{k}^{2} \cdot {\ell}^{-2}}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.5e+80)
    (/
     2.0
     (pow
      (*
       (/ (pow t_m 1.5) l)
       (* (sqrt 2.0) (+ k (* (pow k 3.0) -0.08333333333333333))))
      2.0))
    (/
     2.0
     (/ (* (* 2.0 (sin k)) (* (sin k) (/ (/ (pow t_m 3.0) l) l))) (cos k))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.5e+80) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (sqrt(2.0) * (k + (pow(k, 3.0) * -0.08333333333333333)))), 2.0);
	} else {
		tmp = 2.0 / (((2.0 * sin(k)) * (sin(k) * ((pow(t_m, 3.0) / l) / l))) / cos(k));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.5d+80) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (sqrt(2.0d0) * (k + ((k ** 3.0d0) * (-0.08333333333333333d0))))) ** 2.0d0)
    else
        tmp = 2.0d0 / (((2.0d0 * sin(k)) * (sin(k) * (((t_m ** 3.0d0) / l) / l))) / cos(k))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.5e+80) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.sqrt(2.0) * (k + (Math.pow(k, 3.0) * -0.08333333333333333)))), 2.0);
	} else {
		tmp = 2.0 / (((2.0 * Math.sin(k)) * (Math.sin(k) * ((Math.pow(t_m, 3.0) / l) / l))) / Math.cos(k));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if l <= 1.5e+80:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (math.sqrt(2.0) * (k + (math.pow(k, 3.0) * -0.08333333333333333)))), 2.0)
	else:
		tmp = 2.0 / (((2.0 * math.sin(k)) * (math.sin(k) * ((math.pow(t_m, 3.0) / l) / l))) / math.cos(k))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.5e+80)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(sqrt(2.0) * Float64(k + Float64((k ^ 3.0) * -0.08333333333333333)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * sin(k)) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))) / cos(k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (l <= 1.5e+80)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (sqrt(2.0) * (k + ((k ^ 3.0) * -0.08333333333333333)))) ^ 2.0);
	else
		tmp = 2.0 / (((2.0 * sin(k)) * (sin(k) * (((t_m ^ 3.0) / l) / l))) / cos(k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 1.5e+80], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(k + N[(N[Power[k, 3.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.49999999999999993e80

   1. Initial program 58.0%

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

      1. *-commutative 58.0%

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

      2. sqr-neg 58.0%

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

      3. *-commutative 58.0%

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

      4. associate-*l* 58.0%

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

      5. *-commutative 58.0%

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

      6. sqr-neg 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in k around 0 56.1%

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

      1. add-sqr-sqrt 31.3%

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

      2. pow2 31.3%

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

   7. Applied egg-rr 21.7%

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

      1. *-commutative 21.7%

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

   9. Simplified 21.7%

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

   10. Taylor expanded in k around 0 31.7%

       \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left({k}^{3} \cdot \sqrt{2}\right) + k \cdot \sqrt{2}\right)}\right)}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 31.7%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k \cdot \sqrt{2} + -0.08333333333333333 \cdot \left({k}^{3} \cdot \sqrt{2}\right)\right)}\right)}^{2}} \]

       2. associate-*r* 31.7%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2} + \color{blue}{\left(-0.08333333333333333 \cdot {k}^{3}\right) \cdot \sqrt{2}}\right)\right)}^{2}} \]

       3. distribute-rgt-out 31.7%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(k + -0.08333333333333333 \cdot {k}^{3}\right)\right)}\right)}^{2}} \]

       4. *-commutative 31.7%

          \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + \color{blue}{{k}^{3} \cdot -0.08333333333333333}\right)\right)\right)}^{2}} \]

   12. Simplified 31.7%

       \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)}\right)}^{2}} \]

   if 1.49999999999999993e80 < l

   1. Initial program 36.1%

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

      1. associate-*l* 36.2%

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

      2. sqr-neg 36.2%

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

      3. sqr-neg 36.2%

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

      4. associate-/r* 45.7%

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

      5. distribute-rgt-in 45.7%

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

      6. unpow2 45.7%

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

      7. times-frac 34.6%

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

      8. sqr-neg 34.6%

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

      9. times-frac 45.7%

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

      10. unpow2 45.7%

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

      11. distribute-rgt-in 45.7%

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

   3. Simplified 45.7%

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

      1. cube-mult 45.7%

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

      2. *-un-lft-identity 45.7%

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

      3. times-frac 51.2%

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

      4. pow2 51.2%

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

   6. Applied egg-rr 51.2%

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

   7. Taylor expanded in t around inf 48.6%

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

      1. associate-*r/ 48.6%

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

   9. Simplified 48.6%

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

       1. associate-*r/ 48.6%

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

       2. frac-times 44.8%

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

       3. unpow2 44.8%

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

       4. cube-mult 44.8%

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

       5. *-un-lft-identity 44.8%

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

   11. Applied egg-rr 44.8%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot \sin k\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{\sin k}{\frac{\cos k}{t_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.9e-65)
    (/ 2.0 (* (/ (pow k 3.0) (pow l 2.0)) (/ (sin k) (/ (cos k) t_m))))
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.9e-65) {
		tmp = 2.0 / ((pow(k, 3.0) / pow(l, 2.0)) * (sin(k) / (cos(k) / t_m)));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6.9d-65) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l ** 2.0d0)) * (sin(k) / (cos(k) / t_m)))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.9e-65) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / Math.pow(l, 2.0)) * (Math.sin(k) / (Math.cos(k) / t_m)));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6.9e-65:
		tmp = 2.0 / ((math.pow(k, 3.0) / math.pow(l, 2.0)) * (math.sin(k) / (math.cos(k) / t_m)))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.9e-65)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / (l ^ 2.0)) * Float64(sin(k) / Float64(cos(k) / t_m))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6.9e-65)
		tmp = 2.0 / (((k ^ 3.0) / (l ^ 2.0)) * (sin(k) / (cos(k) / t_m)));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.9e-65], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.89999999999999991e-65

   1. Initial program 49.2%

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

      1. *-commutative 49.2%

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

      2. sqr-neg 49.2%

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

      3. *-commutative 49.2%

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

      4. associate-*l* 49.2%

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

      5. *-commutative 49.2%

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

      6. sqr-neg 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in k around 0 44.8%

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

   6. Taylor expanded in k around inf 56.5%

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

      1. times-frac 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-/l* 57.9%

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

   8. Simplified 57.9%

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

   if 6.89999999999999991e-65 < t

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

      2. sqr-neg 66.1%

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

      3. *-commutative 66.1%

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

      4. associate-*l* 66.0%

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

      5. *-commutative 66.0%

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

      6. sqr-neg 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in k around 0 61.3%

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

      1. add-sqr-sqrt 51.8%

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

      2. pow2 51.8%

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

   7. Applied egg-rr 53.3%

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

      1. *-commutative 53.3%

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

   9. Simplified 53.3%

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

   10. Taylor expanded in k around 0 73.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{{\ell}^{2}} \cdot \frac{\sin k}{\frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t_m \cdot \sqrt[3]{{k}^{2} \cdot {\ell}^{-2}}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.45e+27)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (pow (/ 1.0 (* t_m (cbrt (* (pow k 2.0) (pow 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 tmp;
	if (k <= 1.45e+27) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = pow((1.0 / (t_m * cbrt((pow(k, 2.0) * pow(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 tmp;
	if (k <= 1.45e+27) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = Math.pow((1.0 / (t_m * Math.cbrt((Math.pow(k, 2.0) * Math.pow(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)
	tmp = 0.0
	if (k <= 1.45e+27)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(t_m * cbrt(Float64((k ^ 2.0) * (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_] := N[(t$95$s * If[LessEqual[k, 1.45e+27], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(t$95$m * N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4500000000000001e27

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. sqr-neg 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-*l* 55.8%

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

      5. *-commutative 55.8%

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

      6. sqr-neg 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

   7. Applied egg-rr 22.9%

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

      1. *-commutative 22.9%

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

   9. Simplified 22.9%

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

   10. Taylor expanded in k around 0 28.5%

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

   if 1.4500000000000001e27 < k

   1. Initial program 46.7%

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

      1. *-commutative 46.7%

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

      2. sqr-neg 46.7%

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

      3. *-commutative 46.7%

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

      4. associate-*l* 46.8%

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

      5. *-commutative 46.8%

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

      6. sqr-neg 46.8%

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

   3. Simplified 46.8%

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

      1. associate-/r* 51.1%

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

      2. div-inv 51.1%

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

      3. add-cube-cbrt 50.9%

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

      4. associate-*l* 51.0%

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

      5. pow2 51.0%

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

      6. cbrt-div 51.0%

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

      7. rem-cbrt-cube 51.0%

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

      8. cbrt-div 51.0%

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

      9. rem-cbrt-cube 71.0%

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

   6. Applied egg-rr 71.0%

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

   7. Taylor expanded in k around 0 39.7%

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

      1. associate-/l* 41.2%

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

      2. associate-/r/ 41.3%

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

   9. Simplified 41.3%

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

       1. add-cube-cbrt 41.3%

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

       2. pow3 41.3%

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

   11. Applied egg-rr 57.6%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t \cdot \sqrt[3]{{k}^{2} \cdot {\ell}^{-2}}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t_m \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.16e+27)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (/ 2.0 (* (pow (* t_m (cbrt (/ k (pow l 2.0)))) 3.0) (* 2.0 k))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.16e+27) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / (pow((t_m * cbrt((k / pow(l, 2.0)))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.16e+27) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.cbrt((k / Math.pow(l, 2.0)))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.16e+27)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * cbrt(Float64(k / (l ^ 2.0)))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.16e+27], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.16e27

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

      2. sqr-neg 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-*l* 55.8%

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

      5. *-commutative 55.8%

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

      6. sqr-neg 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

   7. Applied egg-rr 22.9%

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

      1. *-commutative 22.9%

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

   9. Simplified 22.9%

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

   10. Taylor expanded in k around 0 28.5%

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

   if 1.16e27 < k

   1. Initial program 46.7%

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

      1. *-commutative 46.7%

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

      2. sqr-neg 46.7%

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

      3. *-commutative 46.7%

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

      4. associate-*l* 46.8%

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

      5. *-commutative 46.8%

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

      6. sqr-neg 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in k around 0 38.8%

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

   6. Taylor expanded in k around 0 43.2%

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

      1. associate-/l* 43.2%

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

   8. Simplified 43.2%

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

      1. add-cube-cbrt 43.2%

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

      2. pow3 43.2%

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

      3. associate-/r/ 43.2%

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

      4. cbrt-prod 43.2%

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

      5. unpow3 43.2%

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

      6. add-cbrt-cube 59.3%

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

   10. Applied egg-rr 59.3%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.4e-242)
    (/ 2.0 (* (* 2.0 k) (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.4e-242) {
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6.4d-242) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((sin(k) * ((t_m ** 3.0d0) / l)) / l))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.4e-242) {
		tmp = 2.0 / ((2.0 * k) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6.4e-242:
		tmp = 2.0 / ((2.0 * k) * ((math.sin(k) * (math.pow(t_m, 3.0) / l)) / l))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.4e-242)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6.4e-242)
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * ((t_m ^ 3.0) / l)) / l));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.4e-242], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.39999999999999997e-242

   1. Initial program 50.9%

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

      1. *-commutative 50.9%

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

      2. sqr-neg 50.9%

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

      3. *-commutative 50.9%

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

      4. associate-*l* 51.0%

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

      5. *-commutative 51.0%

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

      6. sqr-neg 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in k around 0 50.9%

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

      1. associate-/r* 56.8%

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

      2. associate-*l/ 59.3%

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

   7. Applied egg-rr 59.4%

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

   if 6.39999999999999997e-242 < t

   1. Initial program 57.0%

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

      1. *-commutative 57.0%

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

      2. sqr-neg 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*l* 57.0%

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

      5. *-commutative 57.0%

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

      6. sqr-neg 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in k around 0 50.6%

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

      1. add-sqr-sqrt 45.8%

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

      2. pow2 45.8%

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

   7. Applied egg-rr 47.7%

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

      1. *-commutative 47.7%

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

   9. Simplified 47.7%

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

   10. Taylor expanded in k around 0 64.1%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left(t_m \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.45e+32)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (/ (/ 1.0 k) (pow (* t_m (cbrt (* k (pow 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 tmp;
	if (k <= 2.45e+32) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = (1.0 / k) / pow((t_m * cbrt((k * pow(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 tmp;
	if (k <= 2.45e+32) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = (1.0 / k) / Math.pow((t_m * Math.cbrt((k * Math.pow(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)
	tmp = 0.0
	if (k <= 2.45e+32)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(Float64(1.0 / k) / (Float64(t_m * cbrt(Float64(k * (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_] := N[(t$95$s * If[LessEqual[k, 2.45e+32], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000001e32

   1. Initial program 56.3%

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

      1. *-commutative 56.3%

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

      2. sqr-neg 56.3%

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

      3. *-commutative 56.3%

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

      4. associate-*l* 56.3%

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

      5. *-commutative 56.3%

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

      6. sqr-neg 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. add-sqr-sqrt 27.1%

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

      2. pow2 27.1%

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

   7. Applied egg-rr 23.2%

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

      1. *-commutative 23.2%

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

   9. Simplified 23.2%

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

   10. Taylor expanded in k around 0 28.8%

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

   if 2.4500000000000001e32 < k

   1. Initial program 45.1%

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

      1. *-commutative 45.1%

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

      2. sqr-neg 45.1%

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

      3. *-commutative 45.1%

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

      4. associate-*l* 45.2%

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

      5. *-commutative 45.2%

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

      6. sqr-neg 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in k around 0 38.4%

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

   6. Taylor expanded in k around 0 42.9%

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

      1. associate-/l* 43.0%

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

   8. Simplified 43.0%

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

      1. expm1-log1p-u 39.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)\right)} \]

      2. expm1-udef 41.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)} - 1} \]

      3. associate-/r/ 41.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)}\right)} - 1 \]

      4. *-commutative 41.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \color{blue}{\left(k \cdot 2\right)}}\right)} - 1 \]

   10. Applied egg-rr 41.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 39.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

       2. expm1-log1p 43.0%

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

       3. *-commutative 43.0%

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

       4. associate-/r* 43.0%

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

       5. *-commutative 43.0%

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

       6. associate-/r* 43.0%

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

       7. metadata-eval 43.0%

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

       8. *-commutative 43.0%

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

   12. Simplified 43.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   13. Step-by-step derivation

       1. add-cube-cbrt 43.0%

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

       2. pow3 43.0%

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

       3. cbrt-prod 43.0%

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

       4. unpow3 43.0%

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

       5. add-cbrt-cube 59.5%

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

       6. div-inv 59.5%

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

       7. pow-flip 59.6%

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

       8. metadata-eval 59.6%

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

   14. Applied egg-rr 59.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{\left(t \cdot \sqrt[3]{k \cdot {\ell}^{-2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t_m}^{3} \cdot \left({k}^{2} \cdot {\ell}^{-2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.4e+148)
    (/ 2.0 (* (* 2.0 k) (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
    (/ 1.0 (* (pow t_m 3.0) (* (pow k 2.0) (pow l -2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.4e+148) {
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 1.0 / (pow(t_m, 3.0) * (pow(k, 2.0) * pow(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 (k <= 4.4d+148) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((sin(k) * ((t_m ** 3.0d0) / l)) / l))
    else
        tmp = 1.0d0 / ((t_m ** 3.0d0) * ((k ** 2.0d0) * (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 (k <= 4.4e+148) {
		tmp = 2.0 / ((2.0 * k) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 1.0 / (Math.pow(t_m, 3.0) * (Math.pow(k, 2.0) * Math.pow(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 k <= 4.4e+148:
		tmp = 2.0 / ((2.0 * k) * ((math.sin(k) * (math.pow(t_m, 3.0) / l)) / l))
	else:
		tmp = 1.0 / (math.pow(t_m, 3.0) * (math.pow(k, 2.0) * math.pow(l, -2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.4e+148)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	else
		tmp = Float64(1.0 / Float64((t_m ^ 3.0) * Float64((k ^ 2.0) * (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 (k <= 4.4e+148)
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * ((t_m ^ 3.0) / l)) / l));
	else
		tmp = 1.0 / ((t_m ^ 3.0) * ((k ^ 2.0) * (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[k, 4.4e+148], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.3999999999999998e148

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

      2. sqr-neg 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l* 55.0%

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

      5. *-commutative 55.0%

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

      6. sqr-neg 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0 53.0%

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

      1. associate-/r* 64.1%

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

      2. associate-*l/ 66.7%

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

   7. Applied egg-rr 63.1%

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

   if 4.3999999999999998e148 < k

   1. Initial program 45.3%

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

      1. *-commutative 45.3%

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

      2. sqr-neg 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-*l* 45.3%

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

      5. *-commutative 45.3%

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

      6. sqr-neg 45.3%

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

   3. Simplified 45.3%

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

      1. associate-/r* 45.3%

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

      2. div-inv 45.3%

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

      3. add-cube-cbrt 45.3%

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

      4. associate-*l* 45.3%

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

      5. pow2 45.3%

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

      6. cbrt-div 45.3%

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

      7. rem-cbrt-cube 45.3%

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

      8. cbrt-div 45.3%

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

      9. rem-cbrt-cube 67.0%

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

   6. Applied egg-rr 67.0%

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

   7. Taylor expanded in k around 0 45.2%

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

      1. associate-/l* 45.2%

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

      2. associate-/r/ 45.2%

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

   9. Simplified 45.2%

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

       1. expm1-log1p-u 45.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot {t}^{3}\right)}\right)\right)} \]

       2. expm1-udef 45.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot {t}^{3}\right)}\right)} - 1} \]

       3. associate-/r* 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{{k}^{2}}{{\ell}^{2}} \cdot {t}^{3}}}\right)} - 1 \]

       4. metadata-eval 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{{k}^{2}}{{\ell}^{2}} \cdot {t}^{3}}\right)} - 1 \]

       5. *-commutative 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{t}^{3} \cdot \frac{{k}^{2}}{{\ell}^{2}}}}\right)} - 1 \]

       6. div-inv 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{t}^{3} \cdot \color{blue}{\left({k}^{2} \cdot \frac{1}{{\ell}^{2}}\right)}}\right)} - 1 \]

       7. pow-flip 45.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{t}^{3} \cdot \left({k}^{2} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)}\right)} - 1 \]

       8. metadata-eval 45.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{t}^{3} \cdot \left({k}^{2} \cdot {\ell}^{\color{blue}{-2}}\right)}\right)} - 1 \]

   11. Applied egg-rr 45.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{t}^{3} \cdot \left({k}^{2} \cdot {\ell}^{-2}\right)}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 45.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{t}^{3} \cdot \left({k}^{2} \cdot {\ell}^{-2}\right)}\right)\right)} \]

       2. expm1-log1p 45.3%

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

   13. Simplified 45.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t}^{3} \cdot \left({k}^{2} \cdot {\ell}^{-2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 10^{-244}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t_m}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t_m}^{3}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-244)
    (/ 2.0 (* (* (sin k) (/ (pow t_m 3.0) (* l l))) (* 2.0 k)))
    (/ 2.0 (* (* 2.0 k) (/ (* k (pow t_m 3.0)) (pow l 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-244) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / (l * l))) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * pow(t_m, 3.0)) / pow(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 <= 1d-244) then
        tmp = 2.0d0 / ((sin(k) * ((t_m ** 3.0d0) / (l * l))) * (2.0d0 * k))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * ((k * (t_m ** 3.0d0)) / (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 <= 1e-244) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))) * (2.0 * k));
	} else {
		tmp = 2.0 / ((2.0 * k) * ((k * Math.pow(t_m, 3.0)) / Math.pow(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 <= 1e-244:
		tmp = 2.0 / ((math.sin(k) * (math.pow(t_m, 3.0) / (l * l))) * (2.0 * k))
	else:
		tmp = 2.0 / ((2.0 * k) * ((k * math.pow(t_m, 3.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-244)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(k * (t_m ^ 3.0)) / (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 <= 1e-244)
		tmp = 2.0 / ((sin(k) * ((t_m ^ 3.0) / (l * l))) * (2.0 * k));
	else
		tmp = 2.0 / ((2.0 * k) * ((k * (t_m ^ 3.0)) / (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, 1e-244], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999993e-245

   1. Initial program 50.9%

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

      1. *-commutative 50.9%

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

      2. sqr-neg 50.9%

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

      3. *-commutative 50.9%

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

      4. associate-*l* 51.0%

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

      5. *-commutative 51.0%

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

      6. sqr-neg 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in k around 0 50.9%

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

   if 9.9999999999999993e-245 < t

   1. Initial program 57.0%

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

      1. *-commutative 57.0%

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

      2. sqr-neg 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*l* 57.0%

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

      5. *-commutative 57.0%

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

      6. sqr-neg 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in k around 0 53.0%

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

   6. Taylor expanded in k around 0 53.5%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-244}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \left({t_m}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.45e+148)
    (/ 2.0 (* (* 2.0 k) (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
    (/ 1.0 (* k (* (pow t_m 3.0) (* k (pow l -2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.45e+148) {
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 1.0 / (k * (pow(t_m, 3.0) * (k * pow(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 (k <= 1.45d+148) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((sin(k) * ((t_m ** 3.0d0) / l)) / l))
    else
        tmp = 1.0d0 / (k * ((t_m ** 3.0d0) * (k * (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 (k <= 1.45e+148) {
		tmp = 2.0 / ((2.0 * k) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = 1.0 / (k * (Math.pow(t_m, 3.0) * (k * Math.pow(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 k <= 1.45e+148:
		tmp = 2.0 / ((2.0 * k) * ((math.sin(k) * (math.pow(t_m, 3.0) / l)) / l))
	else:
		tmp = 1.0 / (k * (math.pow(t_m, 3.0) * (k * math.pow(l, -2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.45e+148)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	else
		tmp = Float64(1.0 / Float64(k * Float64((t_m ^ 3.0) * Float64(k * (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 (k <= 1.45e+148)
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * ((t_m ^ 3.0) / l)) / l));
	else
		tmp = 1.0 / (k * ((t_m ^ 3.0) * (k * (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[k, 1.45e+148], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e148

   1. Initial program 54.9%

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

      1. *-commutative 54.9%

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

      2. sqr-neg 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-*l* 55.0%

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

      5. *-commutative 55.0%

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

      6. sqr-neg 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in k around 0 53.0%

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

      1. associate-/r* 64.1%

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

      2. associate-*l/ 66.7%

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

   7. Applied egg-rr 63.1%

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

   if 1.45e148 < k

   1. Initial program 45.3%

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

      1. *-commutative 45.3%

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

      2. sqr-neg 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-*l* 45.3%

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

      5. *-commutative 45.3%

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

      6. sqr-neg 45.3%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in k around 0 39.3%

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

   6. Taylor expanded in k around 0 48.6%

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

      1. associate-/l* 48.6%

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

   8. Simplified 48.6%

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

      1. expm1-log1p-u 45.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)\right)} \]

      2. expm1-udef 48.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)} - 1} \]

      3. associate-/r/ 47.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)}\right)} - 1 \]

      4. *-commutative 47.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \color{blue}{\left(k \cdot 2\right)}}\right)} - 1 \]

   10. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 45.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

       2. expm1-log1p 48.4%

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

       3. *-commutative 48.4%

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

       4. associate-/r* 48.4%

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

       5. *-commutative 48.4%

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

       6. associate-/r* 48.4%

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

       7. metadata-eval 48.4%

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

       8. *-commutative 48.4%

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

   12. Simplified 48.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 45.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)\right)} \]

       2. expm1-udef 47.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)} - 1} \]

       3. div-inv 47.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \color{blue}{\left(k \cdot \frac{1}{{\ell}^{2}}\right)}}\right)} - 1 \]

       4. pow-flip 48.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)}\right)} - 1 \]

       5. metadata-eval 48.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{\color{blue}{-2}}\right)}\right)} - 1 \]

   14. Applied egg-rr 48.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 45.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}\right)\right)} \]

       2. expm1-log1p 48.5%

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

       3. associate-/r* 48.5%

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

   16. Simplified 48.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((k / pow(l, 2.0)) * (pow(t_m, 3.0) * (2.0 * k))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((k / (l ** 2.0d0)) * ((t_m ** 3.0d0) * (2.0d0 * k))))
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 / ((k / Math.pow(l, 2.0)) * (Math.pow(t_m, 3.0) * (2.0 * k))));
}

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 / ((k / math.pow(l, 2.0)) * (math.pow(t_m, 3.0) * (2.0 * k))))

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(k / (l ^ 2.0)) * Float64((t_m ^ 3.0) * Float64(2.0 * k)))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((k / (l ^ 2.0)) * ((t_m ^ 3.0) * (2.0 * k))));
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[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. *-commutative 53.3%

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

   2. sqr-neg 53.3%

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

   3. *-commutative 53.3%

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

   4. associate-*l* 53.4%

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

   5. *-commutative 53.4%

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

   6. sqr-neg 53.4%

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

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 50.8%

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

6. Taylor expanded in k around 0 51.3%

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

   1. associate-/l* 51.0%

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

8. Simplified 51.0%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)\right)} \]

   2. expm1-udef 39.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)} - 1} \]

   3. associate-/r/ 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)}\right)} - 1 \]

   4. *-commutative 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \color{blue}{\left(k \cdot 2\right)}}\right)} - 1 \]

10. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 38.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

    2. expm1-log1p 51.3%

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

    3. associate-*l* 51.2%

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

12. Simplified 51.2%

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

13. Final simplification 51.2%

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

Alternative 19: 55.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / ((2.0 * k) * ((k * pow(t_m, 3.0)) / pow(l, 2.0))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. *-commutative 53.3%

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

   2. sqr-neg 53.3%

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

   3. *-commutative 53.3%

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

   4. associate-*l* 53.4%

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

   5. *-commutative 53.4%

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

   6. sqr-neg 53.4%

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

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 49.2%

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

6. Taylor expanded in k around 0 51.3%

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

7. Final simplification 51.3%

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

Alternative 20: 55.0% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 1.0 (* k (* (pow t_m 3.0) (* k (pow l -2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (1.0 / (k * (pow(t_m, 3.0) * (k * 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 * (1.0d0 / (k * ((t_m ** 3.0d0) * (k * (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 * (1.0 / (k * (Math.pow(t_m, 3.0) * (k * 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 * (1.0 / (k * (math.pow(t_m, 3.0) * (k * 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(1.0 / Float64(k * Float64((t_m ^ 3.0) * Float64(k * (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 * (1.0 / (k * ((t_m ^ 3.0) * (k * (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[(1.0 / N[(k * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. *-commutative 53.3%

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

   2. sqr-neg 53.3%

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

   3. *-commutative 53.3%

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

   4. associate-*l* 53.4%

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

   5. *-commutative 53.4%

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

   6. sqr-neg 53.4%

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

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 50.8%

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

6. Taylor expanded in k around 0 51.3%

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

   1. associate-/l* 51.0%

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

8. Simplified 51.0%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)\right)} \]

   2. expm1-udef 39.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)} - 1} \]

   3. associate-/r/ 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)}\right)} - 1 \]

   4. *-commutative 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \color{blue}{\left(k \cdot 2\right)}}\right)} - 1 \]

10. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 38.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

    2. expm1-log1p 51.3%

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

    3. *-commutative 51.3%

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

    4. associate-/r* 51.0%

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

    5. *-commutative 51.0%

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

    6. associate-/r* 51.0%

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

    7. metadata-eval 51.0%

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

    8. *-commutative 51.0%

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

12. Simplified 51.0%

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 38.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)\right)} \]

    2. expm1-udef 39.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\right)} - 1} \]

    3. div-inv 38.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \color{blue}{\left(k \cdot \frac{1}{{\ell}^{2}}\right)}}\right)} - 1 \]

    4. pow-flip 38.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)}\right)} - 1 \]

    5. metadata-eval 38.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{\color{blue}{-2}}\right)}\right)} - 1 \]

14. Applied egg-rr 38.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 37.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}\right)\right)} \]

    2. expm1-log1p 50.6%

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

    3. associate-/r* 50.9%

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

16. Simplified 50.9%

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

17. Final simplification 50.9%

    \[\leadsto \frac{1}{k \cdot \left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right)} \]
18. Add Preprocessing

Alternative 21: 55.0% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 1.0 k) (* (pow t_m 3.0) (/ k (pow l 2.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((1.0 / k) / (pow(t_m, 3.0) * (k / 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 * ((1.0d0 / k) / ((t_m ** 3.0d0) * (k / (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 * ((1.0 / k) / (Math.pow(t_m, 3.0) * (k / 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 * ((1.0 / k) / (math.pow(t_m, 3.0) * (k / 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(Float64(1.0 / k) / Float64((t_m ^ 3.0) * Float64(k / (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 * ((1.0 / k) / ((t_m ^ 3.0) * (k / (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[(N[(1.0 / k), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. *-commutative 53.3%

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

   2. sqr-neg 53.3%

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

   3. *-commutative 53.3%

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

   4. associate-*l* 53.4%

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

   5. *-commutative 53.4%

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

   6. sqr-neg 53.4%

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

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 50.8%

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

6. Taylor expanded in k around 0 51.3%

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

   1. associate-/l* 51.0%

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

8. Simplified 51.0%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)\right)} \]

   2. expm1-udef 39.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)} - 1} \]

   3. associate-/r/ 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)}\right)} - 1 \]

   4. *-commutative 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \color{blue}{\left(k \cdot 2\right)}}\right)} - 1 \]

10. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 38.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

    2. expm1-log1p 51.3%

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

    3. *-commutative 51.3%

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

    4. associate-/r* 51.0%

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

    5. *-commutative 51.0%

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

    6. associate-/r* 51.0%

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

    7. metadata-eval 51.0%

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

    8. *-commutative 51.0%

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

12. Simplified 51.0%

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

13. Final simplification 51.0%

    \[\leadsto \frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}} \]
14. Add Preprocessing

Alternative 22: 55.0% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 1.0 k) (/ (* k (pow t_m 3.0)) (pow l 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((1.0 / k) / ((k * pow(t_m, 3.0)) / pow(l, 2.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. *-commutative 53.3%

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

   2. sqr-neg 53.3%

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

   3. *-commutative 53.3%

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

   4. associate-*l* 53.4%

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

   5. *-commutative 53.4%

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

   6. sqr-neg 53.4%

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

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 50.8%

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

6. Taylor expanded in k around 0 51.3%

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

   1. associate-/l* 51.0%

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

8. Simplified 51.0%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)\right)} \]

   2. expm1-udef 39.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(2 \cdot k\right)}\right)} - 1} \]

   3. associate-/r/ 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(2 \cdot k\right)}\right)} - 1 \]

   4. *-commutative 39.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \color{blue}{\left(k \cdot 2\right)}}\right)} - 1 \]

10. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 38.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

    2. expm1-log1p 51.3%

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

    3. *-commutative 51.3%

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

    4. associate-/r* 51.0%

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

    5. *-commutative 51.0%

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

    6. associate-/r* 51.0%

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

    7. metadata-eval 51.0%

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

    8. *-commutative 51.0%

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

12. Simplified 51.0%

    \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
13. Step-by-step derivation

    1. associate-*r/ 51.1%

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

14. Applied egg-rr 51.1%

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

15. Final simplification 51.1%

    \[\leadsto \frac{\frac{1}{k}}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))