Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 81.6%

Time: 22.3s

Alternatives: 14

Speedup: 2.0×

• 27.2% 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: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.8 \cdot 10^{-206}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{{l_m}^{2}}}\\ \mathbf{elif}\;t_m \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{{t_m}^{1.5}}{l_m} \cdot \sqrt{\sin k \cdot \tan k}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t_m}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}{\frac{2}{{\left(\frac{t_m}{\sqrt{l_m}}\right)}^{2}}} \cdot \left(\sin k \cdot \frac{t_m}{l_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.8e-206)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (cos k)) (/ (pow (sin k) 2.0) (pow l_m 2.0))))
    (if (<= t_m 3.6e-119)
      (pow
       (/
        (/ (sqrt 2.0) (* (/ (pow t_m 1.5) l_m) (sqrt (* (sin k) (tan k)))))
        (hypot 1.0 (hypot 1.0 (/ k t_m))))
       2.0)
      (/
       1.0
       (*
        (/
         (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
         (/ 2.0 (pow (/ t_m (sqrt l_m)) 2.0)))
        (* (sin k) (/ t_m l_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.8e-206) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * (pow(sin(k), 2.0) / pow(l_m, 2.0)));
	} else if (t_m <= 3.6e-119) {
		tmp = pow(((sqrt(2.0) / ((pow(t_m, 1.5) / l_m) * sqrt((sin(k) * tan(k))))) / hypot(1.0, hypot(1.0, (k / t_m)))), 2.0);
	} else {
		tmp = 1.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) / (2.0 / pow((t_m / sqrt(l_m)), 2.0))) * (sin(k) * (t_m / l_m)));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 6.8e-206) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / Math.pow(l_m, 2.0)));
	} else if (t_m <= 3.6e-119) {
		tmp = Math.pow(((Math.sqrt(2.0) / ((Math.pow(t_m, 1.5) / l_m) * Math.sqrt((Math.sin(k) * Math.tan(k))))) / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)))), 2.0);
	} else {
		tmp = 1.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) / (2.0 / Math.pow((t_m / Math.sqrt(l_m)), 2.0))) * (Math.sin(k) * (t_m / l_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 6.8e-206:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.cos(k)) * (math.pow(math.sin(k), 2.0) / math.pow(l_m, 2.0)))
	elif t_m <= 3.6e-119:
		tmp = math.pow(((math.sqrt(2.0) / ((math.pow(t_m, 1.5) / l_m) * math.sqrt((math.sin(k) * math.tan(k))))) / math.hypot(1.0, math.hypot(1.0, (k / t_m)))), 2.0)
	else:
		tmp = 1.0 / (((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) / (2.0 / math.pow((t_m / math.sqrt(l_m)), 2.0))) * (math.sin(k) * (t_m / l_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 6.8e-206)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * Float64((sin(k) ^ 2.0) / (l_m ^ 2.0))));
	elseif (t_m <= 3.6e-119)
		tmp = Float64(Float64(sqrt(2.0) / Float64(Float64((t_m ^ 1.5) / l_m) * sqrt(Float64(sin(k) * tan(k))))) / hypot(1.0, hypot(1.0, Float64(k / t_m)))) ^ 2.0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / Float64(2.0 / (Float64(t_m / sqrt(l_m)) ^ 2.0))) * Float64(sin(k) * Float64(t_m / l_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 6.8e-206)
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / cos(k)) * ((sin(k) ^ 2.0) / (l_m ^ 2.0)));
	elseif (t_m <= 3.6e-119)
		tmp = ((sqrt(2.0) / (((t_m ^ 1.5) / l_m) * sqrt((sin(k) * tan(k))))) / hypot(1.0, hypot(1.0, (k / t_m)))) ^ 2.0;
	else
		tmp = 1.0 / (((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) / (2.0 / ((t_m / sqrt(l_m)) ^ 2.0))) * (sin(k) * (t_m / l_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-206], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e-119], N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(1.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[Power[N[(t$95$m / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.7999999999999997e-206

   1. Initial program 43.5%

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

      1. associate-*l* 43.5%

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

      2. sqr-neg 43.5%

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

      3. sqr-neg 43.5%

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

      4. associate-/r* 49.8%

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

      5. distribute-rgt-in 49.8%

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

      6. unpow2 49.8%

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

      7. times-frac 37.5%

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

      8. sqr-neg 37.5%

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

      9. times-frac 49.8%

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

      10. unpow2 49.8%

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

      11. distribute-rgt-in 49.8%

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

   3. Simplified 49.8%

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

      1. add-cube-cbrt 49.7%

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

      2. pow3 49.7%

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

      3. associate-/r* 43.4%

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

      4. cbrt-div 43.4%

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

      5. rem-cbrt-cube 49.3%

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

      6. cbrt-prod 64.4%

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

      7. pow2 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in t around 0 50.5%

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

      1. associate-*r* 50.5%

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

      2. *-commutative 50.5%

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

      3. times-frac 53.5%

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

   9. Simplified 53.5%

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

   if 6.7999999999999997e-206 < t < 3.6e-119

   1. Initial program 37.3%

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

      1. associate-/r* 37.3%

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

      2. sqr-neg 37.3%

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

      3. associate-*l* 37.3%

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

      4. sqr-neg 37.3%

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

      5. associate-/r* 41.9%

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

      6. associate-+l+ 41.9%

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

      7. unpow2 41.9%

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

      8. times-frac 23.3%

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

      9. sqr-neg 23.3%

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

      10. times-frac 41.9%

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

      11. unpow2 41.9%

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

   3. Simplified 41.9%

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

      1. add-sqr-sqrt 41.9%

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

      2. pow2 41.9%

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

   6. Applied egg-rr 81.1%

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

   if 3.6e-119 < t

   1. Initial program 64.9%

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

      1. associate-*l* 64.9%

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

      2. sqr-neg 64.9%

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

      3. sqr-neg 64.9%

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

      4. associate-/r* 71.8%

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

      5. distribute-rgt-in 71.8%

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

      6. unpow2 71.8%

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

      7. times-frac 62.9%

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

      8. sqr-neg 62.9%

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

      9. times-frac 71.8%

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

      10. unpow2 71.8%

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

      11. distribute-rgt-in 71.8%

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

   3. Simplified 71.8%

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

      1. associate-/r* 64.9%

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

      2. unpow3 64.9%

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

      3. times-frac 74.6%

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

      4. pow2 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. add-sqr-sqrt 41.6%

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

      2. sqrt-div 41.6%

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

      3. unpow2 41.6%

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

      4. sqrt-prod 41.6%

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

      5. add-sqr-sqrt 41.6%

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

      6. sqrt-div 41.6%

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

      7. unpow2 41.6%

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

      8. sqrt-prod 47.1%

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

      9. add-sqr-sqrt 47.1%

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

   8. Applied egg-rr 47.1%

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

      1. unpow2 47.1%

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

   10. Simplified 47.1%

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

       1. *-un-lft-identity 47.1%

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

       2. associate-/r* 46.7%

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

       3. associate-*l* 49.5%

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

   12. Applied egg-rr 49.5%

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

       1. clear-num 49.0%

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

       2. inv-pow 49.0%

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

       3. associate-/r* 49.0%

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

   14. Applied egg-rr 49.0%

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

       1. unpow-1 49.0%

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

       2. associate-/r/ 50.9%

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

       3. *-commutative 50.9%

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

   16. Simplified 50.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-206}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{2}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{2}{{\left(\frac{t}{\sqrt{\ell}}\right)}^{2}}} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{l_m \cdot l_m}\right)\right) \cdot \left(1 + \left(1 + t_2\right)\right) \leq 10^{+223}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + t_2\right)\right) \cdot \left(\sin k \cdot \left(\frac{t_m}{l_m} \cdot \frac{{t_m}^{2}}{l_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{l_m} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l_m l_m))))
          (+ 1.0 (+ 1.0 t_2)))
         1e+223)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 t_2))
        (* (sin k) (* (/ t_m l_m) (/ (pow t_m 2.0) l_m)))))
      (/
       2.0
       (pow
        (*
         (/ (pow t_m 1.5) l_m)
         (* (sqrt 2.0) (+ k (* (pow k 3.0) -0.08333333333333333))))
        2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2))) <= 1e+223) {
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m / l_m) * (pow(t_m, 2.0) / l_m))));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l_m) * (sqrt(2.0) * (k + (pow(k, 3.0) * -0.08333333333333333)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (((tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l_m * l_m)))) * (1.0d0 + (1.0d0 + t_2))) <= 1d+223) then
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + t_2)) * (sin(k) * ((t_m / l_m) * ((t_m ** 2.0d0) / l_m))))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l_m) * (sqrt(2.0d0) * (k + ((k ** 3.0d0) * (-0.08333333333333333d0))))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2))) <= 1e+223) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + t_2)) * (Math.sin(k) * ((t_m / l_m) * (Math.pow(t_m, 2.0) / l_m))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l_m) * (Math.sqrt(2.0) * (k + (Math.pow(k, 3.0) * -0.08333333333333333)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if ((math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2))) <= 1e+223:
		tmp = 2.0 / ((math.tan(k) * (2.0 + t_2)) * (math.sin(k) * ((t_m / l_m) * (math.pow(t_m, 2.0) / l_m))))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l_m) * (math.sqrt(2.0) * (k + (math.pow(k, 3.0) * -0.08333333333333333)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l_m * l_m)))) * Float64(1.0 + Float64(1.0 + t_2))) <= 1e+223)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_2)) * Float64(sin(k) * Float64(Float64(t_m / l_m) * Float64((t_m ^ 2.0) / l_m)))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l_m) * Float64(sqrt(2.0) * Float64(k + Float64((k ^ 3.0) * -0.08333333333333333)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (((tan(k) * (sin(k) * ((t_m ^ 3.0) / (l_m * l_m)))) * (1.0 + (1.0 + t_2))) <= 1e+223)
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) * (sin(k) * ((t_m / l_m) * ((t_m ^ 2.0) / l_m))));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l_m) * (sqrt(2.0) * (k + ((k ^ 3.0) * -0.08333333333333333)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+223], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l$95$m), $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)) < 1.00000000000000005e223

   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* 88.6%

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

      5. distribute-rgt-in 88.6%

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

      6. unpow2 88.6%

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

      7. times-frac 72.1%

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

      8. sqr-neg 72.1%

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

      9. times-frac 88.6%

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

      10. unpow2 88.6%

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

      11. distribute-rgt-in 88.6%

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

   3. Simplified 88.6%

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

      1. associate-/r* 83.7%

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

      2. unpow3 83.7%

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

      3. times-frac 93.1%

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

      4. pow2 93.1%

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

   6. Applied egg-rr 93.1%

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

   if 1.00000000000000005e223 < (*.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 25.0%

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

      1. *-commutative 25.0%

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

      2. sqr-neg 25.0%

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

      3. *-commutative 25.0%

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

      4. associate-*l* 25.0%

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

      5. *-commutative 25.0%

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

      6. sqr-neg 25.0%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in k around 0 27.6%

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

      1. *-commutative 27.6%

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

   7. Simplified 27.6%

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

      1. add-sqr-sqrt 19.5%

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

      2. pow2 19.5%

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

   9. Applied egg-rr 20.5%

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

       1. *-commutative 20.5%

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

   11. Simplified 20.5%

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

   12. Taylor expanded in k around 0 34.0%

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

       1. associate-*r* 34.0%

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

       2. distribute-rgt-out 34.0%

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

       3. *-commutative 34.0%

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

   14. Simplified 34.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq 10^{+223}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.4 \cdot 10^{-119}:\\ \;\;\;\;2 \cdot \left(\frac{{l_m}^{2}}{t_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t_m}{\sqrt{l_m}}\right)}^{2}}}{\sin k \cdot \frac{t_m}{l_m}} \cdot \frac{1}{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.4e-119)
    (*
     2.0
     (* (/ (pow l_m 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
    (*
     (/ (/ 2.0 (pow (/ t_m (sqrt l_m)) 2.0)) (* (sin k) (/ t_m l_m)))
     (/ 1.0 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 4.4e-119) {
		tmp = 2.0 * ((pow(l_m, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = ((2.0 / pow((t_m / sqrt(l_m)), 2.0)) / (sin(k) * (t_m / l_m))) * (1.0 / (tan(k) * (2.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.4d-119) then
        tmp = 2.0d0 * (((l_m ** 2.0d0) / (t_m * (k ** 2.0d0))) * (cos(k) / (sin(k) ** 2.0d0)))
    else
        tmp = ((2.0d0 / ((t_m / sqrt(l_m)) ** 2.0d0)) / (sin(k) * (t_m / l_m))) * (1.0d0 / (tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 4.4e-119) {
		tmp = 2.0 * ((Math.pow(l_m, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = ((2.0 / Math.pow((t_m / Math.sqrt(l_m)), 2.0)) / (Math.sin(k) * (t_m / l_m))) * (1.0 / (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 4.4e-119:
		tmp = 2.0 * ((math.pow(l_m, 2.0) / (t_m * math.pow(k, 2.0))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = ((2.0 / math.pow((t_m / math.sqrt(l_m)), 2.0)) / (math.sin(k) * (t_m / l_m))) * (1.0 / (math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 4.4e-119)
		tmp = Float64(2.0 * Float64(Float64((l_m ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / (Float64(t_m / sqrt(l_m)) ^ 2.0)) / Float64(sin(k) * Float64(t_m / l_m))) * Float64(1.0 / Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 4.4e-119)
		tmp = 2.0 * (((l_m ^ 2.0) / (t_m * (k ^ 2.0))) * (cos(k) / (sin(k) ^ 2.0)));
	else
		tmp = ((2.0 / ((t_m / sqrt(l_m)) ^ 2.0)) / (sin(k) * (t_m / l_m))) * (1.0 / (tan(k) * (2.0 + ((k / t_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.4e-119], N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[N[(t$95$m / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.4000000000000001e-119

   1. Initial program 42.6%

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

      1. associate-/r* 43.0%

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

      2. sqr-neg 43.0%

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

      3. associate-*l* 39.7%

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

      4. sqr-neg 39.7%

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

      5. associate-/r* 45.8%

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

      6. associate-+l+ 45.8%

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

      7. unpow2 45.8%

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

      8. times-frac 32.7%

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

      9. sqr-neg 32.7%

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

      10. times-frac 45.8%

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

      11. unpow2 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 53.1%

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

      1. associate-*r* 53.1%

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

      2. times-frac 56.2%

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

   7. Simplified 56.2%

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

   if 4.4000000000000001e-119 < t

   1. Initial program 64.9%

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

      1. associate-*l* 64.9%

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

      2. sqr-neg 64.9%

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

      3. sqr-neg 64.9%

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

      4. associate-/r* 71.8%

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

      5. distribute-rgt-in 71.8%

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

      6. unpow2 71.8%

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

      7. times-frac 62.9%

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

      8. sqr-neg 62.9%

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

      9. times-frac 71.8%

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

      10. unpow2 71.8%

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

      11. distribute-rgt-in 71.8%

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

   3. Simplified 71.8%

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

      1. associate-/r* 64.9%

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

      2. unpow3 64.9%

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

      3. times-frac 74.6%

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

      4. pow2 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. add-sqr-sqrt 41.6%

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

      2. sqrt-div 41.6%

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

      3. unpow2 41.6%

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

      4. sqrt-prod 41.6%

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

      5. add-sqr-sqrt 41.6%

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

      6. sqrt-div 41.6%

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

      7. unpow2 41.6%

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

      8. sqrt-prod 47.1%

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

      9. add-sqr-sqrt 47.1%

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

   8. Applied egg-rr 47.1%

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

      1. unpow2 47.1%

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

   10. Simplified 47.1%

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

       1. *-un-lft-identity 47.1%

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

       2. associate-/r* 46.7%

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

       3. associate-*l* 49.5%

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

   12. Applied egg-rr 49.5%

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

       1. div-inv 49.4%

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

       2. associate-/r* 49.6%

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

   14. Applied egg-rr 49.6%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-119}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\sqrt{\ell}}\right)}^{2}}}{\sin k \cdot \frac{t}{\ell}} \cdot \frac{1}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.4 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left(\frac{{l_m}^{2}}{t_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}{\frac{2}{{\left(\frac{t_m}{\sqrt{l_m}}\right)}^{2}}} \cdot \left(\sin k \cdot \frac{t_m}{l_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-137)
    (*
     2.0
     (* (/ (pow l_m 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
    (/
     1.0
     (*
      (/
       (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ 2.0 (pow (/ t_m (sqrt l_m)) 2.0)))
      (* (sin k) (/ t_m l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.4e-137) {
		tmp = 2.0 * ((pow(l_m, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = 1.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) / (2.0 / pow((t_m / sqrt(l_m)), 2.0))) * (sin(k) * (t_m / l_m)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.4d-137) then
        tmp = 2.0d0 * (((l_m ** 2.0d0) / (t_m * (k ** 2.0d0))) * (cos(k) / (sin(k) ** 2.0d0)))
    else
        tmp = 1.0d0 / (((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) / (2.0d0 / ((t_m / sqrt(l_m)) ** 2.0d0))) * (sin(k) * (t_m / l_m)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.4e-137) {
		tmp = 2.0 * ((Math.pow(l_m, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 1.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) / (2.0 / Math.pow((t_m / Math.sqrt(l_m)), 2.0))) * (Math.sin(k) * (t_m / l_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 2.4e-137:
		tmp = 2.0 * ((math.pow(l_m, 2.0) / (t_m * math.pow(k, 2.0))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = 1.0 / (((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) / (2.0 / math.pow((t_m / math.sqrt(l_m)), 2.0))) * (math.sin(k) * (t_m / l_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.4e-137)
		tmp = Float64(2.0 * Float64(Float64((l_m ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / Float64(2.0 / (Float64(t_m / sqrt(l_m)) ^ 2.0))) * Float64(sin(k) * Float64(t_m / l_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 2.4e-137)
		tmp = 2.0 * (((l_m ^ 2.0) / (t_m * (k ^ 2.0))) * (cos(k) / (sin(k) ^ 2.0)));
	else
		tmp = 1.0 / (((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) / (2.0 / ((t_m / sqrt(l_m)) ^ 2.0))) * (sin(k) * (t_m / l_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-137], N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / N[Power[N[(t$95$m / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.4e-137

   1. Initial program 42.5%

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

      1. associate-/r* 42.9%

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

      2. sqr-neg 42.9%

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

      3. associate-*l* 39.6%

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

      4. sqr-neg 39.6%

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

      5. associate-/r* 45.8%

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

      6. associate-+l+ 45.8%

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

      7. unpow2 45.8%

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

      8. times-frac 32.5%

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

      9. sqr-neg 32.5%

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

      10. times-frac 45.8%

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

      11. unpow2 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 53.1%

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

      1. associate-*r* 53.1%

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

      2. times-frac 56.3%

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

   7. Simplified 56.3%

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

   if 2.4e-137 < t

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. sqr-neg 64.6%

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

      3. sqr-neg 64.6%

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

      4. associate-/r* 71.3%

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

      5. distribute-rgt-in 71.3%

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

      6. unpow2 71.3%

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

      7. times-frac 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 71.3%

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

      10. unpow2 71.3%

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

      11. distribute-rgt-in 71.3%

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

   3. Simplified 71.3%

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

      1. associate-/r* 64.6%

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

      2. unpow3 64.6%

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

      3. times-frac 74.2%

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

      4. pow2 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. add-sqr-sqrt 41.7%

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

      2. sqrt-div 41.7%

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

      3. unpow2 41.7%

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

      4. sqrt-prod 41.7%

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

      5. add-sqr-sqrt 41.7%

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

      6. sqrt-div 41.7%

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

      7. unpow2 41.7%

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

      8. sqrt-prod 47.2%

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

      9. add-sqr-sqrt 47.2%

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

   8. Applied egg-rr 47.2%

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

      1. unpow2 47.2%

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

   10. Simplified 47.2%

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

       1. *-un-lft-identity 47.2%

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

       2. associate-/r* 46.7%

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

       3. associate-*l* 49.5%

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

   12. Applied egg-rr 49.5%

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

       1. clear-num 49.0%

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

       2. inv-pow 49.0%

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

       3. associate-/r* 49.0%

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

   14. Applied egg-rr 49.0%

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

       1. unpow-1 49.0%

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

       2. associate-/r/ 50.9%

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

       3. *-commutative 50.9%

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

   16. Simplified 50.9%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{2}{{\left(\frac{t}{\sqrt{\ell}}\right)}^{2}}} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.2 \cdot 10^{-119}:\\ \;\;\;\;2 \cdot \left(\frac{{l_m}^{2}}{t_m \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t_m}{\sqrt{l_m}}\right)}^{2} \cdot \left(\sin k \cdot \frac{t_m}{l_m}\right)}}{\tan k \cdot \left(2 + \frac{k}{t_m} \cdot \frac{k}{t_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-119)
    (*
     2.0
     (* (/ (pow l_m 2.0) (* t_m (pow k 2.0))) (/ (cos k) (pow (sin k) 2.0))))
    (/
     (/ 2.0 (* (pow (/ t_m (sqrt l_m)) 2.0) (* (sin k) (/ t_m l_m))))
     (* (tan k) (+ 2.0 (* (/ k t_m) (/ k t_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.2e-119) {
		tmp = 2.0 * ((pow(l_m, 2.0) / (t_m * pow(k, 2.0))) * (cos(k) / pow(sin(k), 2.0)));
	} else {
		tmp = (2.0 / (pow((t_m / sqrt(l_m)), 2.0) * (sin(k) * (t_m / l_m)))) / (tan(k) * (2.0 + ((k / t_m) * (k / t_m))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7.2d-119) then
        tmp = 2.0d0 * (((l_m ** 2.0d0) / (t_m * (k ** 2.0d0))) * (cos(k) / (sin(k) ** 2.0d0)))
    else
        tmp = (2.0d0 / (((t_m / sqrt(l_m)) ** 2.0d0) * (sin(k) * (t_m / l_m)))) / (tan(k) * (2.0d0 + ((k / t_m) * (k / t_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 7.2e-119) {
		tmp = 2.0 * ((Math.pow(l_m, 2.0) / (t_m * Math.pow(k, 2.0))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (2.0 / (Math.pow((t_m / Math.sqrt(l_m)), 2.0) * (Math.sin(k) * (t_m / l_m)))) / (Math.tan(k) * (2.0 + ((k / t_m) * (k / t_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 7.2e-119:
		tmp = 2.0 * ((math.pow(l_m, 2.0) / (t_m * math.pow(k, 2.0))) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (2.0 / (math.pow((t_m / math.sqrt(l_m)), 2.0) * (math.sin(k) * (t_m / l_m)))) / (math.tan(k) * (2.0 + ((k / t_m) * (k / t_m))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 7.2e-119)
		tmp = Float64(2.0 * Float64(Float64((l_m ^ 2.0) / Float64(t_m * (k ^ 2.0))) * Float64(cos(k) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(2.0 / Float64((Float64(t_m / sqrt(l_m)) ^ 2.0) * Float64(sin(k) * Float64(t_m / l_m)))) / Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 7.2e-119)
		tmp = 2.0 * (((l_m ^ 2.0) / (t_m * (k ^ 2.0))) * (cos(k) / (sin(k) ^ 2.0)));
	else
		tmp = (2.0 / (((t_m / sqrt(l_m)) ^ 2.0) * (sin(k) * (t_m / l_m)))) / (tan(k) * (2.0 + ((k / t_m) * (k / t_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-119], N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[(t$95$m / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e-119

   1. Initial program 42.6%

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

      1. associate-/r* 43.0%

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

      2. sqr-neg 43.0%

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

      3. associate-*l* 39.7%

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

      4. sqr-neg 39.7%

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

      5. associate-/r* 45.8%

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

      6. associate-+l+ 45.8%

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

      7. unpow2 45.8%

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

      8. times-frac 32.7%

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

      9. sqr-neg 32.7%

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

      10. times-frac 45.8%

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

      11. unpow2 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 53.1%

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

      1. associate-*r* 53.1%

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

      2. times-frac 56.2%

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

   7. Simplified 56.2%

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

   if 7.2e-119 < t

   1. Initial program 64.9%

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

      1. associate-*l* 64.9%

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

      2. sqr-neg 64.9%

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

      3. sqr-neg 64.9%

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

      4. associate-/r* 71.8%

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

      5. distribute-rgt-in 71.8%

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

      6. unpow2 71.8%

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

      7. times-frac 62.9%

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

      8. sqr-neg 62.9%

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

      9. times-frac 71.8%

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

      10. unpow2 71.8%

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

      11. distribute-rgt-in 71.8%

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

   3. Simplified 71.8%

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

      1. associate-/r* 64.9%

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

      2. unpow3 64.9%

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

      3. times-frac 74.6%

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

      4. pow2 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. add-sqr-sqrt 41.6%

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

      2. sqrt-div 41.6%

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

      3. unpow2 41.6%

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

      4. sqrt-prod 41.6%

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

      5. add-sqr-sqrt 41.6%

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

      6. sqrt-div 41.6%

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

      7. unpow2 41.6%

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

      8. sqrt-prod 47.1%

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

      9. add-sqr-sqrt 47.1%

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

   8. Applied egg-rr 47.1%

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

      1. unpow2 47.1%

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

   10. Simplified 47.1%

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

       1. *-un-lft-identity 47.1%

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

       2. associate-/r* 46.7%

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

       3. associate-*l* 49.5%

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

   12. Applied egg-rr 49.5%

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

       1. unpow2 71.7%

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

   14. Applied egg-rr 49.5%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-119}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\sqrt{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.65 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{l_m} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \mathbf{elif}\;t_m \leq 2.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t_m}^{3}}{l_m}\right)}{l_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t_m}{\sqrt{l_m}}\right)}^{2} \cdot \left(\sin k \cdot \frac{t_m}{l_m}\right)}}{2 \cdot k}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.65e-42)
    (/
     2.0
     (pow
      (*
       (/ (pow t_m 1.5) l_m)
       (* (sqrt 2.0) (+ k (* (pow k 3.0) -0.08333333333333333))))
      2.0))
    (if (<= t_m 2.5e+91)
      (/
       2.0
       (/
        (*
         (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
         (* (sin k) (/ (pow t_m 3.0) l_m)))
        l_m))
      (/
       (/ 2.0 (* (pow (/ t_m (sqrt l_m)) 2.0) (* (sin k) (/ t_m l_m))))
       (* 2.0 k))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.65e-42) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l_m) * (sqrt(2.0) * (k + (pow(k, 3.0) * -0.08333333333333333)))), 2.0);
	} else if (t_m <= 2.5e+91) {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * (pow(t_m, 3.0) / l_m))) / l_m);
	} else {
		tmp = (2.0 / (pow((t_m / sqrt(l_m)), 2.0) * (sin(k) * (t_m / l_m)))) / (2.0 * k);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.65d-42) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l_m) * (sqrt(2.0d0) * (k + ((k ** 3.0d0) * (-0.08333333333333333d0))))) ** 2.0d0)
    else if (t_m <= 2.5d+91) then
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * (sin(k) * ((t_m ** 3.0d0) / l_m))) / l_m)
    else
        tmp = (2.0d0 / (((t_m / sqrt(l_m)) ** 2.0d0) * (sin(k) * (t_m / l_m)))) / (2.0d0 * k)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 2.65e-42) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l_m) * (Math.sqrt(2.0) * (k + (Math.pow(k, 3.0) * -0.08333333333333333)))), 2.0);
	} else if (t_m <= 2.5e+91) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l_m))) / l_m);
	} else {
		tmp = (2.0 / (Math.pow((t_m / Math.sqrt(l_m)), 2.0) * (Math.sin(k) * (t_m / l_m)))) / (2.0 * k);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 2.65e-42:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l_m) * (math.sqrt(2.0) * (k + (math.pow(k, 3.0) * -0.08333333333333333)))), 2.0)
	elif t_m <= 2.5e+91:
		tmp = 2.0 / (((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * (math.sin(k) * (math.pow(t_m, 3.0) / l_m))) / l_m)
	else:
		tmp = (2.0 / (math.pow((t_m / math.sqrt(l_m)), 2.0) * (math.sin(k) * (t_m / l_m)))) / (2.0 * k)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 2.65e-42)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l_m) * Float64(sqrt(2.0) * Float64(k + Float64((k ^ 3.0) * -0.08333333333333333)))) ^ 2.0));
	elseif (t_m <= 2.5e+91)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64((t_m ^ 3.0) / l_m))) / l_m));
	else
		tmp = Float64(Float64(2.0 / Float64((Float64(t_m / sqrt(l_m)) ^ 2.0) * Float64(sin(k) * Float64(t_m / l_m)))) / Float64(2.0 * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 2.65e-42)
		tmp = 2.0 / ((((t_m ^ 1.5) / l_m) * (sqrt(2.0) * (k + ((k ^ 3.0) * -0.08333333333333333)))) ^ 2.0);
	elseif (t_m <= 2.5e+91)
		tmp = 2.0 / (((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * (sin(k) * ((t_m ^ 3.0) / l_m))) / l_m);
	else
		tmp = (2.0 / (((t_m / sqrt(l_m)) ^ 2.0) * (sin(k) * (t_m / l_m)))) / (2.0 * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.65e-42], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l$95$m), $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], If[LessEqual[t$95$m, 2.5e+91], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[(t$95$m / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.65e-42

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. *-commutative 45.4%

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

      4. associate-*l* 45.4%

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

      5. *-commutative 45.4%

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

      6. sqr-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in k around 0 43.8%

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

      1. *-commutative 43.8%

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

   7. Simplified 43.8%

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

      1. add-sqr-sqrt 20.1%

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

      2. pow2 20.1%

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

   9. Applied egg-rr 15.5%

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

       1. *-commutative 15.5%

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

   11. Simplified 15.5%

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

   12. Taylor expanded in k around 0 19.7%

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

       1. associate-*r* 19.7%

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

       2. distribute-rgt-out 19.7%

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

       3. *-commutative 19.7%

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

   14. Simplified 19.7%

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

   if 2.65e-42 < t < 2.5000000000000001e91

   1. Initial program 72.3%

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

      1. associate-*l* 72.3%

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

      2. sqr-neg 72.3%

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

      3. sqr-neg 72.3%

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

      4. associate-/r* 81.0%

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

      5. distribute-rgt-in 81.0%

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

      6. unpow2 81.0%

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

      7. times-frac 75.0%

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

      8. sqr-neg 75.0%

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

      9. times-frac 81.0%

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

      10. unpow2 81.0%

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

      11. distribute-rgt-in 81.0%

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

   3. Simplified 81.0%

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

      1. associate-*l/ 87.0%

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

      2. associate-*l/ 93.1%

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

   6. Applied egg-rr 93.1%

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

   if 2.5000000000000001e91 < t

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. sqr-neg 58.4%

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

      3. sqr-neg 58.4%

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

      4. associate-/r* 64.5%

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

      5. distribute-rgt-in 64.5%

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

      6. unpow2 64.5%

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

      7. times-frac 51.0%

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

      8. sqr-neg 51.0%

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

      9. times-frac 64.5%

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

      10. unpow2 64.5%

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

      11. distribute-rgt-in 64.5%

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

   3. Simplified 64.5%

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

      1. associate-/r* 58.4%

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

      2. unpow3 58.4%

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

      3. times-frac 70.0%

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

      4. pow2 70.0%

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

   6. Applied egg-rr 70.0%

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

      1. add-sqr-sqrt 44.5%

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

      2. sqrt-div 44.5%

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

      3. unpow2 44.5%

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

      4. sqrt-prod 44.5%

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

      5. add-sqr-sqrt 44.5%

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

      6. sqrt-div 44.4%

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

      7. unpow2 44.4%

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

      8. sqrt-prod 55.2%

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

      9. add-sqr-sqrt 55.3%

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

   8. Applied egg-rr 55.3%

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

      1. unpow2 55.3%

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

   10. Simplified 55.3%

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

       1. *-un-lft-identity 55.3%

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

       2. associate-/r* 55.8%

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

       3. associate-*l* 57.6%

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

   12. Applied egg-rr 57.6%

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

   13. Taylor expanded in k around 0 46.6%

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

       1. *-commutative 64.8%

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

   15. Simplified 46.6%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.65 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sqrt{2} \cdot \left(k + {k}^{3} \cdot -0.08333333333333333\right)\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\sqrt{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}}{2 \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t_m}{l_m} \cdot \frac{{t_m}^{2}}{l_m}\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t_m}{\sqrt{l_m}}\right)}^{2} \cdot \left(\sin k \cdot \frac{t_m}{l_m}\right)}}{\tan k \cdot \left(2 + \frac{k}{t_m} \cdot \frac{k}{t_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.9e-211)
    (/
     2.0
     (*
      (* (sin k) (* (/ t_m l_m) (/ (pow t_m 2.0) l_m)))
      (* 2.0 (/ (sin k) (cos k)))))
    (/
     (/ 2.0 (* (pow (/ t_m (sqrt l_m)) 2.0) (* (sin k) (/ t_m l_m))))
     (* (tan k) (+ 2.0 (* (/ k t_m) (/ k t_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.9e-211) {
		tmp = 2.0 / ((sin(k) * ((t_m / l_m) * (pow(t_m, 2.0) / l_m))) * (2.0 * (sin(k) / cos(k))));
	} else {
		tmp = (2.0 / (pow((t_m / sqrt(l_m)), 2.0) * (sin(k) * (t_m / l_m)))) / (tan(k) * (2.0 + ((k / t_m) * (k / t_m))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.9d-211) then
        tmp = 2.0d0 / ((sin(k) * ((t_m / l_m) * ((t_m ** 2.0d0) / l_m))) * (2.0d0 * (sin(k) / cos(k))))
    else
        tmp = (2.0d0 / (((t_m / sqrt(l_m)) ** 2.0d0) * (sin(k) * (t_m / l_m)))) / (tan(k) * (2.0d0 + ((k / t_m) * (k / t_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (t_m <= 1.9e-211) {
		tmp = 2.0 / ((Math.sin(k) * ((t_m / l_m) * (Math.pow(t_m, 2.0) / l_m))) * (2.0 * (Math.sin(k) / Math.cos(k))));
	} else {
		tmp = (2.0 / (Math.pow((t_m / Math.sqrt(l_m)), 2.0) * (Math.sin(k) * (t_m / l_m)))) / (Math.tan(k) * (2.0 + ((k / t_m) * (k / t_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if t_m <= 1.9e-211:
		tmp = 2.0 / ((math.sin(k) * ((t_m / l_m) * (math.pow(t_m, 2.0) / l_m))) * (2.0 * (math.sin(k) / math.cos(k))))
	else:
		tmp = (2.0 / (math.pow((t_m / math.sqrt(l_m)), 2.0) * (math.sin(k) * (t_m / l_m)))) / (math.tan(k) * (2.0 + ((k / t_m) * (k / t_m))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (t_m <= 1.9e-211)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64(t_m / l_m) * Float64((t_m ^ 2.0) / l_m))) * Float64(2.0 * Float64(sin(k) / cos(k)))));
	else
		tmp = Float64(Float64(2.0 / Float64((Float64(t_m / sqrt(l_m)) ^ 2.0) * Float64(sin(k) * Float64(t_m / l_m)))) / Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (t_m <= 1.9e-211)
		tmp = 2.0 / ((sin(k) * ((t_m / l_m) * ((t_m ^ 2.0) / l_m))) * (2.0 * (sin(k) / cos(k))));
	else
		tmp = (2.0 / (((t_m / sqrt(l_m)) ^ 2.0) * (sin(k) * (t_m / l_m)))) / (tan(k) * (2.0 + ((k / t_m) * (k / t_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e-211], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[(t$95$m / N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.90000000000000006e-211

   1. Initial program 43.4%

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

      1. associate-*l* 43.4%

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

      2. sqr-neg 43.4%

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

      3. sqr-neg 43.4%

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

      4. associate-/r* 49.8%

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

      5. distribute-rgt-in 49.8%

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

      6. unpow2 49.8%

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

      7. times-frac 38.1%

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

      8. sqr-neg 38.1%

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

      9. times-frac 49.8%

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

      10. unpow2 49.8%

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

      11. distribute-rgt-in 49.8%

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

   3. Simplified 49.8%

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

      1. associate-/r* 43.4%

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

      2. unpow3 43.4%

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

      3. times-frac 54.9%

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

      4. pow2 54.9%

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

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in t around inf 53.8%

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

   if 1.90000000000000006e-211 < t

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. sqr-neg 59.8%

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

      3. sqr-neg 59.8%

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

      4. associate-/r* 66.2%

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

      5. distribute-rgt-in 66.2%

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

      6. unpow2 66.2%

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

      7. times-frac 55.0%

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

      8. sqr-neg 55.0%

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

      9. times-frac 66.2%

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

      10. unpow2 66.2%

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

      11. distribute-rgt-in 66.2%

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

   3. Simplified 66.2%

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

      1. associate-/r* 59.8%

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

      2. unpow3 59.8%

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

      3. times-frac 69.1%

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

      4. pow2 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. add-sqr-sqrt 37.8%

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

      2. sqrt-div 37.8%

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

      3. unpow2 37.8%

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

      4. sqrt-prod 37.8%

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

      5. add-sqr-sqrt 37.8%

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

      6. sqrt-div 37.8%

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

      7. unpow2 37.8%

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

      8. sqrt-prod 44.0%

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

      9. add-sqr-sqrt 44.0%

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

   8. Applied egg-rr 44.0%

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

      1. unpow2 44.0%

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

   10. Simplified 44.0%

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

       1. *-un-lft-identity 44.0%

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

       2. associate-/r* 44.2%

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

       3. associate-*l* 46.4%

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

   12. Applied egg-rr 46.4%

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

       1. unpow2 66.1%

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

   14. Applied egg-rr 46.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right) \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{t}{\sqrt{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t_m}^{1.5}}{l_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {t_2}^{2}}}{2 + \frac{k}{t_m} \cdot \frac{k}{t_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l_m)))
   (*
    t_s
    (if (<= k 1.3e-24)
      (/ 2.0 (pow (* t_2 (* k (sqrt 2.0))) 2.0))
      (/
       (/ 2.0 (* (* (sin k) (tan k)) (pow t_2 2.0)))
       (+ 2.0 (* (/ k t_m) (/ k t_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(t_m, 1.5) / l_m;
	double tmp;
	if (k <= 1.3e-24) {
		tmp = 2.0 / pow((t_2 * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = (2.0 / ((sin(k) * tan(k)) * pow(t_2, 2.0))) / (2.0 + ((k / t_m) * (k / t_m)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m ** 1.5d0) / l_m
    if (k <= 1.3d-24) then
        tmp = 2.0d0 / ((t_2 * (k * sqrt(2.0d0))) ** 2.0d0)
    else
        tmp = (2.0d0 / ((sin(k) * tan(k)) * (t_2 ** 2.0d0))) / (2.0d0 + ((k / t_m) * (k / t_m)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = Math.pow(t_m, 1.5) / l_m;
	double tmp;
	if (k <= 1.3e-24) {
		tmp = 2.0 / Math.pow((t_2 * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = (2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow(t_2, 2.0))) / (2.0 + ((k / t_m) * (k / t_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	t_2 = math.pow(t_m, 1.5) / l_m
	tmp = 0
	if k <= 1.3e-24:
		tmp = 2.0 / math.pow((t_2 * (k * math.sqrt(2.0))), 2.0)
	else:
		tmp = (2.0 / ((math.sin(k) * math.tan(k)) * math.pow(t_2, 2.0))) / (2.0 + ((k / t_m) * (k / t_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64((t_m ^ 1.5) / l_m)
	tmp = 0.0
	if (k <= 1.3e-24)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (t_2 ^ 2.0))) / Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	t_2 = (t_m ^ 1.5) / l_m;
	tmp = 0.0;
	if (k <= 1.3e-24)
		tmp = 2.0 / ((t_2 * (k * sqrt(2.0))) ^ 2.0);
	else
		tmp = (2.0 / ((sin(k) * tan(k)) * (t_2 ^ 2.0))) / (2.0 + ((k / t_m) * (k / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.3e-24], N[(2.0 / N[Power[N[(t$95$2 * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-24

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. sqr-neg 52.6%

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

      3. *-commutative 52.6%

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

      4. associate-*l* 52.6%

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

      5. *-commutative 52.6%

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

      6. sqr-neg 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in k around 0 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

      1. add-sqr-sqrt 29.2%

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

      2. pow2 29.2%

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

   9. Applied egg-rr 29.3%

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

       1. *-commutative 29.3%

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

   11. Simplified 29.3%

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

   12. Taylor expanded in k around 0 37.6%

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

   if 1.3e-24 < k

   1. Initial program 48.3%

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

      1. associate-/r* 48.7%

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

      2. sqr-neg 48.7%

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

      3. associate-*l* 48.7%

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

      4. sqr-neg 48.7%

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

      5. associate-/r* 56.9%

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

      6. associate-+l+ 56.9%

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

      7. unpow2 56.9%

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

      8. times-frac 47.2%

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

      9. sqr-neg 47.2%

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

      10. times-frac 56.9%

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

      11. unpow2 56.9%

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

   3. Simplified 56.9%

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

      1. add-sqr-sqrt 35.9%

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

      2. pow2 35.9%

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

      3. associate-/r* 30.6%

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

      4. sqrt-div 30.6%

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

      5. sqrt-pow1 33.8%

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

      6. metadata-eval 33.8%

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

      7. sqrt-prod 24.8%

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

      8. add-sqr-sqrt 44.3%

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

   6. Applied egg-rr 44.3%

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

      1. unpow2 56.3%

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

   8. Applied egg-rr 44.3%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + \frac{k}{t} \cdot \frac{k}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t_m}^{1.5}}{l_m} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + \frac{k}{t_m} \cdot \frac{k}{t_m}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{1}{\frac{l_m}{{t_m}^{3}}}}{l_m}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.4e+16)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l_m) (* k (sqrt 2.0))) 2.0))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (* (/ k t_m) (/ k t_m))))
      (* (sin k) (/ (/ 1.0 (/ l_m (pow t_m 3.0))) l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 6.4e+16) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l_m) * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) * (k / t_m)))) * (sin(k) * ((1.0 / (l_m / pow(t_m, 3.0))) / l_m)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.4d+16) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l_m) * (k * sqrt(2.0d0))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) * (k / t_m)))) * (sin(k) * ((1.0d0 / (l_m / (t_m ** 3.0d0))) / l_m)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 6.4e+16) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l_m) * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + ((k / t_m) * (k / t_m)))) * (Math.sin(k) * ((1.0 / (l_m / Math.pow(t_m, 3.0))) / l_m)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 6.4e+16:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l_m) * (k * math.sqrt(2.0))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k) * (2.0 + ((k / t_m) * (k / t_m)))) * (math.sin(k) * ((1.0 / (l_m / math.pow(t_m, 3.0))) / l_m)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 6.4e+16)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l_m) * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + Float64(Float64(k / t_m) * Float64(k / t_m)))) * Float64(sin(k) * Float64(Float64(1.0 / Float64(l_m / (t_m ^ 3.0))) / l_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 6.4e+16)
		tmp = 2.0 / ((((t_m ^ 1.5) / l_m) * (k * sqrt(2.0))) ^ 2.0);
	else
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) * (k / t_m)))) * (sin(k) * ((1.0 / (l_m / (t_m ^ 3.0))) / l_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 6.4e+16], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(1.0 / N[(l$95$m / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.4e16

   1. Initial program 51.0%

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

      1. *-commutative 51.0%

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

      2. sqr-neg 51.0%

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

      3. *-commutative 51.0%

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

      4. associate-*l* 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 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

      1. add-sqr-sqrt 28.4%

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

      2. pow2 28.4%

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

   9. Applied egg-rr 28.5%

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

       1. *-commutative 28.5%

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

   11. Simplified 28.5%

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

   12. Taylor expanded in k around 0 36.5%

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

   if 6.4e16 < k

   1. Initial program 52.8%

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

      1. associate-*l* 52.8%

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

      2. sqr-neg 52.8%

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

      3. sqr-neg 52.8%

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

      4. associate-/r* 60.5%

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

      5. distribute-rgt-in 60.5%

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

      6. unpow2 60.5%

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

      7. times-frac 49.2%

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

      8. sqr-neg 49.2%

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

      9. times-frac 60.5%

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

      10. unpow2 60.5%

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

      11. distribute-rgt-in 60.5%

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

   3. Simplified 60.5%

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

      1. clear-num 60.5%

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

      2. inv-pow 60.5%

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

   6. Applied egg-rr 60.5%

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

      1. unpow-1 60.5%

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

   8. Simplified 60.5%

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

      1. unpow2 60.5%

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

   10. Applied egg-rr 60.5%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{1}{\frac{\ell}{{t}^{3}}}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.92 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{l_m}}{l_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{l_m}^{2}}{{k}^{2}}}{{t_m}^{3}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.92e+30)
    (/ 2.0 (* (* 2.0 k) (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
    (/ (/ (pow l_m 2.0) (pow k 2.0)) (pow t_m 3.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.92e+30) {
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = (pow(l_m, 2.0) / pow(k, 2.0)) / pow(t_m, 3.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.92d+30) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((sin(k) * ((t_m ** 3.0d0) / l_m)) / l_m))
    else
        tmp = ((l_m ** 2.0d0) / (k ** 2.0d0)) / (t_m ** 3.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.92e+30) {
		tmp = 2.0 / ((2.0 * k) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = (Math.pow(l_m, 2.0) / Math.pow(k, 2.0)) / Math.pow(t_m, 3.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.92e+30:
		tmp = 2.0 / ((2.0 * k) * ((math.sin(k) * (math.pow(t_m, 3.0) / l_m)) / l_m))
	else:
		tmp = (math.pow(l_m, 2.0) / math.pow(k, 2.0)) / math.pow(t_m, 3.0)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.92e+30)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	else
		tmp = Float64(Float64((l_m ^ 2.0) / (k ^ 2.0)) / (t_m ^ 3.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.92e+30)
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * ((t_m ^ 3.0) / l_m)) / l_m));
	else
		tmp = ((l_m ^ 2.0) / (k ^ 2.0)) / (t_m ^ 3.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 1.92e+30], 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$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.9200000000000001e30

   1. Initial program 52.4%

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

      1. associate-*l* 52.4%

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

      2. sqr-neg 52.4%

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

      3. sqr-neg 52.4%

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

      4. associate-/r* 58.2%

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

      5. distribute-rgt-in 58.2%

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

      6. unpow2 58.2%

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

      7. times-frac 47.0%

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

      8. sqr-neg 47.0%

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

      9. times-frac 58.2%

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

      10. unpow2 58.2%

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

      11. distribute-rgt-in 58.2%

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

   3. Simplified 58.2%

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

      1. associate-/r* 52.4%

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

      2. unpow3 52.4%

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

      3. times-frac 61.4%

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

      4. pow2 61.4%

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

   6. Applied egg-rr 61.4%

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

   7. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

   9. Simplified 57.6%

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

       1. associate-*r/ 57.5%

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

   11. Applied egg-rr 57.5%

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

       1. associate-*l/ 59.4%

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

       2. associate-*l/ 58.4%

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

       3. unpow2 58.4%

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

       4. unpow3 58.4%

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

   13. Applied egg-rr 58.4%

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

   if 1.9200000000000001e30 < k

   1. Initial program 47.8%

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

      1. associate-*l* 47.8%

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

      2. sqr-neg 47.8%

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

      3. sqr-neg 47.8%

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

      4. associate-/r* 56.3%

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

      5. distribute-rgt-in 56.3%

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

      6. unpow2 56.3%

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

      7. times-frac 43.8%

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

      8. sqr-neg 43.8%

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

      9. times-frac 56.3%

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

      10. unpow2 56.3%

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

      11. distribute-rgt-in 56.3%

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

   3. Simplified 56.3%

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

      1. add-cube-cbrt 56.2%

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

      2. pow3 56.2%

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

      3. associate-/r* 47.7%

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

      4. cbrt-div 47.6%

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

      5. rem-cbrt-cube 55.0%

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

      6. cbrt-prod 71.3%

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

      7. pow2 71.3%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in k around 0 42.0%

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

      1. associate-/r* 42.1%

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

   9. Simplified 42.1%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.92 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 1.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (pow (* (/ (pow t_m 1.5) l_m) (* k (sqrt 2.0))) 2.0))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / pow(((pow(t_m, 1.5) / l_m) * (k * sqrt(2.0))), 2.0));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. *-commutative 51.4%

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

   2. sqr-neg 51.4%

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

   3. *-commutative 51.4%

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

   4. associate-*l* 51.4%

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

   5. *-commutative 51.4%

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

   6. sqr-neg 51.4%

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

3. Simplified 51.4%

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

5. Taylor expanded in k around 0 48.0%

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

   1. *-commutative 48.0%

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

7. Simplified 48.0%

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

   1. add-sqr-sqrt 27.2%

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

   2. pow2 27.2%

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

9. Applied egg-rr 25.6%

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

    1. *-commutative 25.6%

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

11. Simplified 25.6%

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

12. Taylor expanded in k around 0 36.3%

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

13. Final simplification 36.3%

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

Alternative 12: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.32 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{l_m}}{l_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{{l_m}^{2}}{{t_m}^{3}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.32e+30)
    (/ 2.0 (* (* 2.0 k) (/ (* (sin k) (/ (pow t_m 3.0) l_m)) l_m)))
    (/ 2.0 (* (* 2.0 k) (/ k (/ (pow l_m 2.0) (pow t_m 3.0))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.32e+30) {
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * (pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = 2.0 / ((2.0 * k) * (k / (pow(l_m, 2.0) / pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.32d+30) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((sin(k) * ((t_m ** 3.0d0) / l_m)) / l_m))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (k / ((l_m ** 2.0d0) / (t_m ** 3.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.32e+30) {
		tmp = 2.0 / ((2.0 * k) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l_m)) / l_m));
	} else {
		tmp = 2.0 / ((2.0 * k) * (k / (Math.pow(l_m, 2.0) / Math.pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.32e+30:
		tmp = 2.0 / ((2.0 * k) * ((math.sin(k) * (math.pow(t_m, 3.0) / l_m)) / l_m))
	else:
		tmp = 2.0 / ((2.0 * k) * (k / (math.pow(l_m, 2.0) / math.pow(t_m, 3.0))))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.32e+30)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l_m)) / l_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k / Float64((l_m ^ 2.0) / (t_m ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.32e+30)
		tmp = 2.0 / ((2.0 * k) * ((sin(k) * ((t_m ^ 3.0) / l_m)) / l_m));
	else
		tmp = 2.0 / ((2.0 * k) * (k / ((l_m ^ 2.0) / (t_m ^ 3.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * If[LessEqual[k, 2.32e+30], 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$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k / N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.32e30

   1. Initial program 52.4%

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

      1. associate-*l* 52.4%

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

      2. sqr-neg 52.4%

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

      3. sqr-neg 52.4%

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

      4. associate-/r* 58.2%

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

      5. distribute-rgt-in 58.2%

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

      6. unpow2 58.2%

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

      7. times-frac 47.0%

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

      8. sqr-neg 47.0%

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

      9. times-frac 58.2%

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

      10. unpow2 58.2%

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

      11. distribute-rgt-in 58.2%

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

   3. Simplified 58.2%

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

      1. associate-/r* 52.4%

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

      2. unpow3 52.4%

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

      3. times-frac 61.4%

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

      4. pow2 61.4%

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

   6. Applied egg-rr 61.4%

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

   7. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

   9. Simplified 57.6%

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

       1. associate-*r/ 57.5%

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

   11. Applied egg-rr 57.5%

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

       1. associate-*l/ 59.4%

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

       2. associate-*l/ 58.4%

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

       3. unpow2 58.4%

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

       4. unpow3 58.4%

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

   13. Applied egg-rr 58.4%

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

   if 2.32e30 < k

   1. Initial program 47.8%

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

      1. associate-*l* 47.8%

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

      2. sqr-neg 47.8%

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

      3. sqr-neg 47.8%

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

      4. associate-/r* 56.3%

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

      5. distribute-rgt-in 56.3%

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

      6. unpow2 56.3%

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

      7. times-frac 43.8%

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

      8. sqr-neg 43.8%

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

      9. times-frac 56.3%

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

      10. unpow2 56.3%

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

      11. distribute-rgt-in 56.3%

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

   3. Simplified 56.3%

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

      1. associate-/r* 47.8%

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

      2. unpow3 47.8%

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

      3. times-frac 63.2%

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

      4. pow2 63.2%

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

   6. Applied egg-rr 63.2%

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

   7. Taylor expanded in k around 0 39.5%

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

      1. *-commutative 39.5%

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

   9. Simplified 39.5%

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

       1. associate-*r/ 39.5%

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

   11. Applied egg-rr 39.5%

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

   12. Taylor expanded in k around 0 42.9%

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

       1. associate-/l* 42.9%

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

   14. Simplified 42.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.32 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.7% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 k) (/ k (/ (pow l_m 2.0) (pow t_m 3.0)))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / ((2.0 * k) * (k / (pow(l_m, 2.0) / pow(t_m, 3.0)))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k / N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. associate-*l* 51.4%

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

   2. sqr-neg 51.4%

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

   3. sqr-neg 51.4%

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

   4. associate-/r* 57.8%

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

   5. distribute-rgt-in 57.8%

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

   6. unpow2 57.8%

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

   7. times-frac 46.3%

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

   8. sqr-neg 46.3%

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

   9. times-frac 57.8%

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

   10. unpow2 57.8%

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

   11. distribute-rgt-in 57.8%

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

3. Simplified 57.8%

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

   1. associate-/r* 51.4%

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

   2. unpow3 51.4%

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

   3. times-frac 61.8%

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

   4. pow2 61.8%

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

6. Applied egg-rr 61.8%

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

7. Taylor expanded in k around 0 53.6%

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

   1. *-commutative 53.6%

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

9. Simplified 53.6%

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

    1. associate-*r/ 53.6%

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

11. Applied egg-rr 53.6%

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

12. Taylor expanded in k around 0 51.2%

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

    1. associate-/l* 50.8%

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

14. Simplified 50.8%

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

15. Final simplification 50.8%

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

Alternative 14: 55.3% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 k) (/ (* k (pow t_m 3.0)) (pow l_m 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * (2.0 / ((2.0 * k) * ((k * pow(t_m, 3.0)) / pow(l_m, 2.0))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.4%

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

   1. associate-*l* 51.4%

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

   2. sqr-neg 51.4%

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

   3. sqr-neg 51.4%

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

   4. associate-/r* 57.8%

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

   5. distribute-rgt-in 57.8%

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

   6. unpow2 57.8%

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

   7. times-frac 46.3%

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

   8. sqr-neg 46.3%

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

   9. times-frac 57.8%

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

   10. unpow2 57.8%

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

   11. distribute-rgt-in 57.8%

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

3. Simplified 57.8%

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

   1. associate-/r* 51.4%

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

   2. unpow3 51.4%

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

   3. times-frac 61.8%

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

   4. pow2 61.8%

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

6. Applied egg-rr 61.8%

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

7. Taylor expanded in k around 0 53.6%

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

   1. *-commutative 53.6%

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

9. Simplified 53.6%

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

    1. associate-*r/ 53.6%

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

11. Applied egg-rr 53.6%

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

12. Taylor expanded in k around 0 51.2%

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

13. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2024018 
(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))))