Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 90.7%

Time: 22.4s

Alternatives: 24

Speedup: 1.9×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right)\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 1.95e-115)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 1.5e+98)
        (* t_2 (* (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))) t_2))
        (/
         2.0
         (pow
          (*
           t_m
           (*
            (pow (cbrt l) -2.0)
            (* (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))) (cbrt (sin k)))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.95e-115) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 1.5e+98) {
		tmp = t_2 * ((2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * t_2);
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))) * cbrt(sin(k))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.95e-115) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 1.5e+98) {
		tmp = t_2 * ((2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * t_2);
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * Math.cbrt(Math.sin(k))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 1.95e-115)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 1.5e+98)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * t_2));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * cbrt(sin(k))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e-115], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+98], N[(t$95$2 * N[(N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.9499999999999999e-115

   1. Initial program 49.6%

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

   3. Simplified 50.1%

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

      1. add-sqr-sqrt 31.3%

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

      2. pow2 31.3%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in t around 0 32.9%

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

      1. *-commutative 32.9%

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

      2. unpow-prod-down 30.8%

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

      3. pow2 30.8%

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

      4. add-sqr-sqrt 74.0%

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

      5. times-frac 74.0%

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

   9. Applied egg-rr 74.0%

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

       1. times-frac 74.0%

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

       2. associate-*r/ 74.0%

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

   11. Simplified 74.0%

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

   if 1.9499999999999999e-115 < t < 1.5000000000000001e98

   1. Initial program 82.0%

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

   3. Simplified 82.1%

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

      1. associate-*r* 86.4%

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

      2. add-sqr-sqrt 86.2%

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

      3. times-frac 86.5%

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

   6. Applied egg-rr 97.3%

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

      1. associate-/l* 97.4%

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

      2. associate-*l* 97.4%

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

   8. Simplified 97.4%

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

   if 1.5000000000000001e98 < t

   1. Initial program 63.3%

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

   3. Simplified 48.1%

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

      1. add-cube-cbrt 48.1%

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

      2. pow3 48.1%

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

      3. cbrt-div 48.1%

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

      4. rem-cbrt-cube 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

   8. Applied egg-rr 68.1%

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

      1. associate-*l* 68.3%

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

   10. Simplified 68.3%

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

       1. *-commutative 68.3%

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

       2. metadata-eval 68.3%

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

       3. associate-+r+ 68.3%

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

       4. cbrt-prod 95.1%

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

       5. associate-+r+ 95.1%

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

       6. metadata-eval 95.1%

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

   12. Applied egg-rr 95.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;t\_2 \cdot \left(\frac{2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 1.95e-115)
      (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
      (if (<= t_m 4.3e+98)
        (* t_2 (* (/ 2.0 (* (pow t_m 3.0) (* (sin k) (tan k)))) t_2))
        (/
         2.0
         (pow
          (* t_m (* (pow (cbrt l) -2.0) (* (cbrt (sin k)) (cbrt (* k 2.0)))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.95e-115) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.3e+98) {
		tmp = t_2 * ((2.0 / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * t_2);
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt(sin(k)) * cbrt((k * 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.95e-115) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.3e+98) {
		tmp = t_2 * ((2.0 / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * t_2);
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt(Math.sin(k)) * Math.cbrt((k * 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 1.95e-115)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.3e+98)
		tmp = Float64(t_2 * Float64(Float64(2.0 / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * t_2));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(sin(k)) * cbrt(Float64(k * 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e-115], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.3e+98], N[(t$95$2 * N[(N[(2.0 / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.9499999999999999e-115

   1. Initial program 49.6%

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

   3. Simplified 50.1%

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

      1. add-sqr-sqrt 31.3%

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

      2. pow2 31.3%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in t around 0 32.9%

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

      1. *-commutative 32.9%

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

      2. unpow-prod-down 30.8%

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

      3. pow2 30.8%

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

      4. add-sqr-sqrt 74.0%

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

      5. times-frac 74.0%

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

   9. Applied egg-rr 74.0%

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

       1. times-frac 74.0%

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

       2. associate-*r/ 74.0%

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

   11. Simplified 74.0%

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

   if 1.9499999999999999e-115 < t < 4.3000000000000001e98

   1. Initial program 82.0%

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

   3. Simplified 82.1%

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

      1. associate-*r* 86.4%

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

      2. add-sqr-sqrt 86.2%

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

      3. times-frac 86.5%

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

   6. Applied egg-rr 97.3%

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

      1. associate-/l* 97.4%

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

      2. associate-*l* 97.4%

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

   8. Simplified 97.4%

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

   if 4.3000000000000001e98 < t

   1. Initial program 63.3%

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

   3. Simplified 48.1%

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

      1. add-cube-cbrt 48.1%

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

      2. pow3 48.1%

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

      3. cbrt-div 48.1%

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

      4. rem-cbrt-cube 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

   8. Applied egg-rr 68.1%

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

      1. associate-*l* 68.3%

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

   10. Simplified 68.3%

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

       1. *-commutative 68.3%

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

       2. metadata-eval 68.3%

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

       3. associate-+r+ 68.3%

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

       4. cbrt-prod 95.1%

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

       5. associate-+r+ 95.1%

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

       6. metadata-eval 95.1%

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

   12. Applied egg-rr 95.1%

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

   13. Taylor expanded in k around 0 90.7%

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

       1. *-commutative 90.7%

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

   15. Simplified 90.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 125000:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 125000.0)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 125000.0) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 125000.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 125000.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 125000.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 125000

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 125000 < t

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

      1. add-cube-cbrt 70.3%

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

      2. pow3 70.3%

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

      3. *-commutative 70.3%

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

      4. cbrt-prod 70.3%

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

      5. cbrt-div 70.4%

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

      6. rem-cbrt-cube 78.8%

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

      7. cbrt-prod 89.0%

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

      8. pow2 89.0%

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

   6. Applied egg-rr 89.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 125000:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 27000:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.65 \cdot 10^{+106}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot 2}}{t\_m \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+272}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 27000.0)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 2.65e+106)
      (*
       (pow (/ (cbrt (* l 2.0)) (* t_m (cbrt (* (sin k) (tan k))))) 3.0)
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (if (<= t_m 2.5e+272)
        (/
         2.0
         (pow
          (* t_m (* (pow (cbrt l) -2.0) (* (cbrt (sin k)) (cbrt (* k 2.0)))))
          3.0))
        (/
         2.0
         (* (* k 2.0) (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 27000.0) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 2.65e+106) {
		tmp = pow((cbrt((l * 2.0)) / (t_m * cbrt((sin(k) * tan(k))))), 3.0) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else if (t_m <= 2.5e+272) {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt(sin(k)) * cbrt((k * 2.0))))), 3.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 27000.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 2.65e+106) {
		tmp = Math.pow((Math.cbrt((l * 2.0)) / (t_m * Math.cbrt((Math.sin(k) * Math.tan(k))))), 3.0) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else if (t_m <= 2.5e+272) {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt(Math.sin(k)) * Math.cbrt((k * 2.0))))), 3.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 27000.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 2.65e+106)
		tmp = Float64((Float64(cbrt(Float64(l * 2.0)) / Float64(t_m * cbrt(Float64(sin(k) * tan(k))))) ^ 3.0) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	elseif (t_m <= 2.5e+272)
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(sin(k)) * cbrt(Float64(k * 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 27000.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.65e+106], N[(N[Power[N[(N[Power[N[(l * 2.0), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+272], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 27000

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 27000 < t < 2.65e106

   1. Initial program 80.3%

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

   3. Simplified 80.5%

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

      1. associate-*r* 87.8%

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

      2. *-un-lft-identity 87.8%

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

      3. times-frac 87.9%

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

      4. associate-/l/ 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. /-rgt-identity 88.0%

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

      2. associate-*l/ 87.9%

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

      3. associate-*l* 87.8%

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

   8. Simplified 87.8%

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

      1. add-cube-cbrt 87.6%

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

      2. pow3 87.6%

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

      3. cbrt-div 87.5%

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

      4. *-commutative 87.5%

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

      5. cbrt-prod 87.2%

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

      6. unpow3 87.2%

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

      7. add-cbrt-cube 95.3%

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

   10. Applied egg-rr 95.3%

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

   if 2.65e106 < t < 2.49999999999999986e272

   1. Initial program 60.0%

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

   3. Simplified 39.4%

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

      1. add-cube-cbrt 39.4%

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

      2. pow3 39.4%

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

      3. cbrt-div 39.4%

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

      4. rem-cbrt-cube 55.2%

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

   6. Applied egg-rr 55.2%

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

      1. add-cube-cbrt 55.2%

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

      2. pow3 55.2%

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

   8. Applied egg-rr 59.7%

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

      1. associate-*l* 60.0%

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

   10. Simplified 60.0%

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

       1. *-commutative 60.0%

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

       2. metadata-eval 60.0%

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

       3. associate-+r+ 60.0%

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

       4. cbrt-prod 96.4%

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

       5. associate-+r+ 96.4%

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

       6. metadata-eval 96.4%

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

   12. Applied egg-rr 96.4%

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

   13. Taylor expanded in k around 0 90.3%

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

       1. *-commutative 90.3%

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

   15. Simplified 90.3%

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

   if 2.49999999999999986e272 < t

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

      1. add-cube-cbrt 83.3%

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

      2. pow3 83.3%

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

      3. cbrt-div 83.3%

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

      4. rem-cbrt-cube 84.6%

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

      5. cbrt-prod 100.0%

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

      6. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 83.3%

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

   9. Simplified 100.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 27000:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+106}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+272}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.28:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.28)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 4.1e+99)
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ (* l 2.0) (* (tan k) (* (sin k) (pow t_m 3.0)))))
      (/
       2.0
       (pow
        (* t_m (* (pow (cbrt l) -2.0) (* (cbrt (sin k)) (cbrt (* k 2.0)))))
        3.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.28) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.1e+99) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * ((l * 2.0) / (tan(k) * (sin(k) * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * (cbrt(sin(k)) * cbrt((k * 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.28) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.1e+99) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l * 2.0) / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * (Math.cbrt(Math.sin(k)) * Math.cbrt((k * 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.28)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.1e+99)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l * 2.0) / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * Float64(cbrt(sin(k)) * cbrt(Float64(k * 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.28], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e+99], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(k * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 0.28000000000000003

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 0.28000000000000003 < t < 4.09999999999999979e99

   1. Initial program 85.4%

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

   3. Simplified 85.6%

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

      1. associate-*r* 94.6%

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

      2. *-un-lft-identity 94.6%

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

      3. times-frac 94.7%

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

      4. associate-/l/ 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. /-rgt-identity 94.8%

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

      2. associate-*l/ 94.7%

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

      3. associate-*l* 94.7%

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

   8. Simplified 94.7%

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

      1. associate-*r* 94.7%

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

      2. pow1 94.7%

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

      3. *-commutative 94.7%

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

      4. *-commutative 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. unpow1 94.7%

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

       2. *-commutative 94.7%

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

   12. Simplified 94.7%

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

   if 4.09999999999999979e99 < t

   1. Initial program 63.3%

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

   3. Simplified 48.1%

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

      1. add-cube-cbrt 48.1%

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

      2. pow3 48.1%

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

      3. cbrt-div 48.1%

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

      4. rem-cbrt-cube 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

   8. Applied egg-rr 68.1%

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

      1. associate-*l* 68.3%

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

   10. Simplified 68.3%

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

       1. *-commutative 68.3%

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

       2. metadata-eval 68.3%

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

       3. associate-+r+ 68.3%

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

       4. cbrt-prod 95.1%

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

       5. associate-+r+ 95.1%

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

       6. metadata-eval 95.1%

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

   12. Applied egg-rr 95.1%

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

   13. Taylor expanded in k around 0 90.7%

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

       1. *-commutative 90.7%

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

   15. Simplified 90.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.28:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{k \cdot 2}\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 82:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 82.0)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (/
     2.0
     (*
      (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
      (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 82.0) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 82.0d0) then
        tmp = (cos(k) / t_m) * ((l * (sqrt(2.0d0) / (k * sin(k)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((tan(k) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0)))) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 82.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 82.0:
		tmp = (math.cos(k) / t_m) * math.pow((l * (math.sqrt(2.0) / (k * math.sin(k)))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k) * (1.0 + (1.0 + math.pow((k / t_m), 2.0)))) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 82.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 82.0)
		tmp = (cos(k) / t_m) * ((l * (sqrt(2.0) / (k * sin(k)))) ^ 2.0);
	else
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + ((k / t_m) ^ 2.0)))) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 82.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 82

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 82 < t

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

      1. add-sqr-sqrt 70.4%

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

      2. pow2 70.4%

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

      3. sqrt-div 70.4%

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

      4. sqrt-pow1 78.7%

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

      5. metadata-eval 78.7%

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

      6. sqrt-prod 48.8%

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

      7. add-sqr-sqrt 86.3%

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

   6. Applied egg-rr 86.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 82:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.35 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 2.35e+134)
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
        (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (/
       2.0
       (*
        (* k 2.0)
        (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 2.35e+134) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 2.35e+134) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 2.35e+134)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.35e+134], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.05000000000000004

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 1.05000000000000004 < t < 2.35000000000000013e134

   1. Initial program 72.9%

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

   3. Simplified 72.9%

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

      1. unpow3 72.9%

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

      2. times-frac 92.9%

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

      3. pow2 92.9%

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

   6. Applied egg-rr 92.9%

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

   if 2.35000000000000013e134 < t

   1. Initial program 68.3%

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

   3. Simplified 68.3%

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

      1. add-cube-cbrt 68.3%

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

      2. pow3 68.3%

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

      3. *-commutative 68.3%

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

      4. cbrt-prod 68.3%

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

      5. cbrt-div 68.3%

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

      6. rem-cbrt-cube 74.9%

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

      7. cbrt-prod 86.1%

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

      8. pow2 86.1%

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

   6. Applied egg-rr 86.1%

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

   7. Taylor expanded in k around 0 83.3%

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

      1. *-commutative 68.3%

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

   9. Simplified 83.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 280:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 280.0)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 6.2e+135)
      (/
       2.0
       (*
        (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
        (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (/
       2.0
       (* (* k 2.0) (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 280.0) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 6.2e+135) {
		tmp = 2.0 / ((tan(k) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 280.0) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 6.2e+135) {
		tmp = 2.0 / ((Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 280.0)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 6.2e+135)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 280.0], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+135], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 280

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 280 < t < 6.20000000000000044e135

   1. Initial program 72.9%

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

   3. Simplified 72.9%

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

      1. unpow3 72.9%

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

      2. times-frac 92.9%

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

      3. pow2 92.9%

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

   6. Applied egg-rr 92.9%

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

   if 6.20000000000000044e135 < t

   1. Initial program 68.3%

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

   3. Simplified 68.3%

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

      1. add-cube-cbrt 68.3%

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

      2. pow3 68.3%

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

      3. cbrt-div 68.3%

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

      4. rem-cbrt-cube 74.9%

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

      5. cbrt-prod 86.4%

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

      6. pow2 86.4%

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in k around 0 83.5%

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

      1. *-commutative 68.3%

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

   9. Simplified 83.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 280:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t\_m}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 4.1e+99)
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (* l (/ (/ 2.0 (tan k)) (* (sin k) (pow t_m 3.0)))))
      (/
       2.0
       (* (* k 2.0) (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.1e+99) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * (l * ((2.0 / tan(k)) / (sin(k) * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.1e+99) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * (l * ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.1e+99)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e+99], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.3500000000000001

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 1.3500000000000001 < t < 4.09999999999999979e99

   1. Initial program 85.4%

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

   3. Simplified 85.6%

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

      1. associate-*r* 94.6%

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

      2. *-un-lft-identity 94.6%

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

      3. times-frac 94.7%

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

      4. associate-/l/ 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. /-rgt-identity 94.8%

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

      2. *-commutative 94.8%

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

      3. metadata-eval 94.8%

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

      4. distribute-neg-frac 94.8%

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

      5. associate-/l/ 94.7%

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

      6. distribute-neg-frac 94.7%

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

      7. distribute-neg-frac 94.7%

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

      8. metadata-eval 94.7%

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

      9. *-commutative 94.7%

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

   8. Simplified 94.7%

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

   if 4.09999999999999979e99 < t

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-div 63.3%

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

      4. rem-cbrt-cube 75.8%

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

      5. cbrt-prod 87.0%

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

      6. pow2 87.0%

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in k around 0 82.5%

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

      1. *-commutative 63.3%

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

   9. Simplified 82.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.32:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t\_m}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.32)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 4.1e+99)
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (/ (* l 2.0) (* (tan k) (* (sin k) (pow t_m 3.0)))))
      (/
       2.0
       (* (* k 2.0) (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.32) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 4.1e+99) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * ((l * 2.0) / (tan(k) * (sin(k) * pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.32) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 4.1e+99) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l * 2.0) / (Math.tan(k) * (Math.sin(k) * Math.pow(t_m, 3.0))));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.32)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 4.1e+99)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l * 2.0) / Float64(tan(k) * Float64(sin(k) * (t_m ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.32], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e+99], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 0.320000000000000007

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow-prod-down 33.4%

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

      3. pow2 33.4%

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

      4. add-sqr-sqrt 76.1%

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

      5. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. times-frac 76.1%

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

       2. associate-*r/ 76.1%

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

   11. Simplified 76.1%

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

   if 0.320000000000000007 < t < 4.09999999999999979e99

   1. Initial program 85.4%

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

   3. Simplified 85.6%

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

      1. associate-*r* 94.6%

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

      2. *-un-lft-identity 94.6%

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

      3. times-frac 94.7%

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

      4. associate-/l/ 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. /-rgt-identity 94.8%

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

      2. associate-*l/ 94.7%

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

      3. associate-*l* 94.7%

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

   8. Simplified 94.7%

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

      1. associate-*r* 94.7%

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

      2. pow1 94.7%

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

      3. *-commutative 94.7%

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

      4. *-commutative 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. unpow1 94.7%

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

       2. *-commutative 94.7%

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

   12. Simplified 94.7%

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

   if 4.09999999999999979e99 < t

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-div 63.3%

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

      4. rem-cbrt-cube 75.8%

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

      5. cbrt-prod 87.0%

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

      6. pow2 87.0%

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in k around 0 82.5%

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

      1. *-commutative 63.3%

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

   9. Simplified 82.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.32:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell \cdot 2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{\ell \cdot 2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.6e+70)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 1.15e+103)
      (* (/ (* l 2.0) (* (pow t_m 3.0) (* (sin k) (tan k)))) (* l 0.5))
      (if (<= t_m 4e+125)
        (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (pow k 2.0))))
        (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow t_m 3.0) (* l l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.6e+70) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 1.15e+103) {
		tmp = ((l * 2.0) / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * (l * 0.5);
	} else if (t_m <= 4e+125) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * pow(k, 2.0)));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.6e+70) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 1.15e+103) {
		tmp = ((l * 2.0) / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * (l * 0.5);
	} else if (t_m <= 4e+125) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * Math.pow(k, 2.0)));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.6e+70)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 1.15e+103)
		tmp = Float64(Float64(Float64(l * 2.0) / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * Float64(l * 0.5));
	elseif (t_m <= 4e+125)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * (k ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.6e+70], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+103], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+125], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 6.60000000000000033e70

   1. Initial program 55.3%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in t around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow-prod-down 35.3%

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

      3. pow2 35.3%

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

      4. add-sqr-sqrt 77.0%

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

      5. times-frac 77.0%

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

   9. Applied egg-rr 77.0%

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

       1. times-frac 77.0%

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

       2. associate-*r/ 77.0%

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

   11. Simplified 77.0%

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

   if 6.60000000000000033e70 < t < 1.15000000000000004e103

   1. Initial program 83.0%

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

   3. Simplified 83.1%

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

      1. associate-*r* 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.6%

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

      4. associate-/l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-*l/ 99.4%

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

      3. associate-*l* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in k around 0 82.0%

      \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 82.0%

          \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 82.0%

       \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 1.15000000000000004e103 < t < 3.9999999999999997e125

   1. Initial program 1.6%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in k around 0 0.0%

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

      1. add-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

      3. associate-/l/ 0.0%

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

      4. cbrt-div 0.0%

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

      5. unpow3 0.0%

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

      6. add-cbrt-cube 75.0%

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

      7. cbrt-unprod 75.0%

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

      8. unpow2 75.0%

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

      9. div-inv 75.0%

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

      10. unpow-prod-down 0.0%

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

      11. pow-flip 0.0%

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

      12. metadata-eval 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. cube-prod 75.0%

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

   9. Simplified 75.0%

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

   if 3.9999999999999997e125 < t

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. *-commutative 67.2%

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

   7. Simplified 67.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{\ell \cdot 2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e+86)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (/ 2.0 (* (* k 2.0) (* (sin k) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e+86) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * pow((t_m / pow(cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e+86) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e+86)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e+86], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000004e86

   1. Initial program 55.9%

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

   3. Simplified 56.3%

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

      1. add-sqr-sqrt 40.7%

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

      2. pow2 40.7%

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

   6. Applied egg-rr 41.0%

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

   7. Taylor expanded in t around 0 38.2%

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

      1. *-commutative 38.2%

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

      2. unpow-prod-down 36.1%

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

      3. pow2 36.1%

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

      4. add-sqr-sqrt 76.4%

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

      5. times-frac 76.4%

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

   9. Applied egg-rr 76.4%

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

       1. times-frac 76.4%

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

       2. associate-*r/ 76.5%

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

   11. Simplified 76.5%

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

   if 2.80000000000000004e86 < t

   1. Initial program 64.2%

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

   3. Simplified 64.2%

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

      1. add-cube-cbrt 64.2%

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

      2. pow3 64.2%

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

      3. cbrt-div 64.2%

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

      4. rem-cbrt-cube 76.3%

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

      5. cbrt-prod 87.3%

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

      6. pow2 87.3%

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

   6. Applied egg-rr 87.3%

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

   7. Taylor expanded in k around 0 82.9%

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

      1. *-commutative 64.2%

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

   9. Simplified 82.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\cos k}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{\ell \cdot 2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;t\_m \leq 3.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e+70)
    (* (/ (cos k) t_m) (pow (* l (/ (sqrt 2.0) (* k (sin k)))) 2.0))
    (if (<= t_m 1.15e+103)
      (* (/ (* l 2.0) (* (pow t_m 3.0) (* (sin k) (tan k)))) (* l 0.5))
      (if (<= t_m 3.6e+125)
        (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (* (pow t_m 2.0) (/ t_m l)) l)))
        (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow t_m 3.0) (* l l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.2e+70) {
		tmp = (cos(k) / t_m) * pow((l * (sqrt(2.0) / (k * sin(k)))), 2.0);
	} else if (t_m <= 1.15e+103) {
		tmp = ((l * 2.0) / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * (l * 0.5);
	} else if (t_m <= 3.6e+125) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 6.2d+70) then
        tmp = (cos(k) / t_m) * ((l * (sqrt(2.0d0) / (k * sin(k)))) ** 2.0d0)
    else if (t_m <= 1.15d+103) then
        tmp = ((l * 2.0d0) / ((t_m ** 3.0d0) * (sin(k) * tan(k)))) * (l * 0.5d0)
    else if (t_m <= 3.6d+125) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 2.0d0) * (t_m / l)) / l))
    else
        tmp = 2.0d0 / ((k * 2.0d0) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.2e+70) {
		tmp = (Math.cos(k) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k * Math.sin(k)))), 2.0);
	} else if (t_m <= 1.15e+103) {
		tmp = ((l * 2.0) / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * (l * 0.5);
	} else if (t_m <= 3.6e+125) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 6.2e+70:
		tmp = (math.cos(k) / t_m) * math.pow((l * (math.sqrt(2.0) / (k * math.sin(k)))), 2.0)
	elif t_m <= 1.15e+103:
		tmp = ((l * 2.0) / (math.pow(t_m, 3.0) * (math.sin(k) * math.tan(k)))) * (l * 0.5)
	elif t_m <= 3.6e+125:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 2.0) * (t_m / l)) / l))
	else:
		tmp = 2.0 / ((k * 2.0) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.2e+70)
		tmp = Float64(Float64(cos(k) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k * sin(k)))) ^ 2.0));
	elseif (t_m <= 1.15e+103)
		tmp = Float64(Float64(Float64(l * 2.0) / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * Float64(l * 0.5));
	elseif (t_m <= 3.6e+125)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 6.2e+70)
		tmp = (cos(k) / t_m) * ((l * (sqrt(2.0) / (k * sin(k)))) ^ 2.0);
	elseif (t_m <= 1.15e+103)
		tmp = ((l * 2.0) / ((t_m ^ 3.0) * (sin(k) * tan(k)))) * (l * 0.5);
	elseif (t_m <= 3.6e+125)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 2.0) * (t_m / l)) / l));
	else
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e+70], N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+103], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.6e+125], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 6.2000000000000006e70

   1. Initial program 55.3%

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

   3. Simplified 55.7%

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

      1. add-sqr-sqrt 39.7%

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

      2. pow2 39.7%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in t around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow-prod-down 35.3%

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

      3. pow2 35.3%

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

      4. add-sqr-sqrt 77.0%

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

      5. times-frac 77.0%

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

   9. Applied egg-rr 77.0%

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

       1. times-frac 77.0%

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

       2. associate-*r/ 77.0%

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

   11. Simplified 77.0%

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

   if 6.2000000000000006e70 < t < 1.15000000000000004e103

   1. Initial program 83.0%

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

   3. Simplified 83.1%

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

      1. associate-*r* 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. times-frac 99.6%

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

      4. associate-/l/ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. associate-*l/ 99.4%

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

      3. associate-*l* 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in k around 0 82.0%

      \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 82.0%

          \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 82.0%

       \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 1.15000000000000004e103 < t < 3.6000000000000003e125

   1. Initial program 1.6%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in k around 0 0.0%

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

      1. unpow3 0.0%

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

      2. *-un-lft-identity 0.0%

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

      3. times-frac 75.0%

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

      4. pow2 75.0%

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

   7. Applied egg-rr 75.0%

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

   if 3.6000000000000003e125 < t

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. *-commutative 67.2%

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

   7. Simplified 67.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{\ell \cdot 2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.28:\\ \;\;\;\;{\left({t\_m}^{-0.5} \cdot \frac{\ell \cdot \sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\tan k}}{k \cdot {t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.28)
    (pow (* (pow t_m -0.5) (/ (* l (sqrt 2.0)) (pow k 2.0))) 2.0)
    (/
     (* (* l l) (/ (/ 2.0 (tan k)) (* k (pow t_m 3.0))))
     (+ 2.0 (pow (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.28) {
		tmp = pow((pow(t_m, -0.5) * ((l * sqrt(2.0)) / pow(k, 2.0))), 2.0);
	} else {
		tmp = ((l * l) * ((2.0 / tan(k)) / (k * pow(t_m, 3.0)))) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 0.28d0) then
        tmp = ((t_m ** (-0.5d0)) * ((l * sqrt(2.0d0)) / (k ** 2.0d0))) ** 2.0d0
    else
        tmp = ((l * l) * ((2.0d0 / tan(k)) / (k * (t_m ** 3.0d0)))) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.28) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * ((l * Math.sqrt(2.0)) / Math.pow(k, 2.0))), 2.0);
	} else {
		tmp = ((l * l) * ((2.0 / Math.tan(k)) / (k * Math.pow(t_m, 3.0)))) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 0.28:
		tmp = math.pow((math.pow(t_m, -0.5) * ((l * math.sqrt(2.0)) / math.pow(k, 2.0))), 2.0)
	else:
		tmp = ((l * l) * ((2.0 / math.tan(k)) / (k * math.pow(t_m, 3.0)))) / (2.0 + math.pow((k / t_m), 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.28)
		tmp = Float64((t_m ^ -0.5) * Float64(Float64(l * sqrt(2.0)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / tan(k)) / Float64(k * (t_m ^ 3.0)))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 0.28)
		tmp = ((t_m ^ -0.5) * ((l * sqrt(2.0)) / (k ^ 2.0))) ^ 2.0;
	else
		tmp = ((l * l) * ((2.0 / tan(k)) / (k * (t_m ^ 3.0)))) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.28], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 0.28000000000000003

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

   8. Taylor expanded in k around 0 23.8%

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

      1. pow1 23.8%

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

      2. associate-/l* 23.8%

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

      3. pow1/2 23.8%

         \[\leadsto {\left({\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \color{blue}{{\left(\frac{1}{t}\right)}^{0.5}}\right)}^{1}\right)}^{2} \]

      4. inv-pow 23.8%

         \[\leadsto {\left({\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot {\color{blue}{\left({t}^{-1}\right)}}^{0.5}\right)}^{1}\right)}^{2} \]

      5. pow-pow 23.8%

         \[\leadsto {\left({\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \color{blue}{{t}^{\left(-1 \cdot 0.5\right)}}\right)}^{1}\right)}^{2} \]

      6. metadata-eval 23.8%

         \[\leadsto {\left({\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot {t}^{\color{blue}{-0.5}}\right)}^{1}\right)}^{2} \]

   10. Applied egg-rr 23.8%

       \[\leadsto {\color{blue}{\left({\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot {t}^{-0.5}\right)}^{1}\right)}}^{2} \]
   11. Step-by-step derivation

       1. unpow1 23.8%

          \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot {t}^{-0.5}\right)}}^{2} \]

       2. associate-*r/ 23.8%

          \[\leadsto {\left(\color{blue}{\frac{\ell \cdot \sqrt{2}}{{k}^{2}}} \cdot {t}^{-0.5}\right)}^{2} \]

   12. Simplified 23.8%

       \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot {t}^{-0.5}\right)}}^{2} \]

   if 0.28000000000000003 < t

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in k around 0 66.0%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.28:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\ell \cdot \sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\tan k}}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.3:\\ \;\;\;\;{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t\_m}^{-0.5}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\tan k}}{k \cdot {t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.3)
    (pow (/ (* (* l (sqrt 2.0)) (pow t_m -0.5)) (pow k 2.0)) 2.0)
    (/
     (* (* l l) (/ (/ 2.0 (tan k)) (* k (pow t_m 3.0))))
     (+ 2.0 (pow (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.3) {
		tmp = pow((((l * sqrt(2.0)) * pow(t_m, -0.5)) / pow(k, 2.0)), 2.0);
	} else {
		tmp = ((l * l) * ((2.0 / tan(k)) / (k * pow(t_m, 3.0)))) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 0.3d0) then
        tmp = (((l * sqrt(2.0d0)) * (t_m ** (-0.5d0))) / (k ** 2.0d0)) ** 2.0d0
    else
        tmp = ((l * l) * ((2.0d0 / tan(k)) / (k * (t_m ** 3.0d0)))) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 0.3) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) * Math.pow(t_m, -0.5)) / Math.pow(k, 2.0)), 2.0);
	} else {
		tmp = ((l * l) * ((2.0 / Math.tan(k)) / (k * Math.pow(t_m, 3.0)))) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 0.3:
		tmp = math.pow((((l * math.sqrt(2.0)) * math.pow(t_m, -0.5)) / math.pow(k, 2.0)), 2.0)
	else:
		tmp = ((l * l) * ((2.0 / math.tan(k)) / (k * math.pow(t_m, 3.0)))) / (2.0 + math.pow((k / t_m), 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 0.3)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) * (t_m ^ -0.5)) / (k ^ 2.0)) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(Float64(2.0 / tan(k)) / Float64(k * (t_m ^ 3.0)))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 0.3)
		tmp = (((l * sqrt(2.0)) * (t_m ^ -0.5)) / (k ^ 2.0)) ^ 2.0;
	else
		tmp = ((l * l) * ((2.0 / tan(k)) / (k * (t_m ^ 3.0)))) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 0.3], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, -0.5], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 0.299999999999999989

   1. Initial program 53.1%

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

   3. Simplified 53.5%

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

      1. add-sqr-sqrt 37.0%

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

      2. pow2 37.0%

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in t around 0 35.7%

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

   8. Taylor expanded in k around 0 23.8%

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

      1. associate-*l/ 23.8%

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

      2. pow1/2 23.8%

         \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{{\left(\frac{1}{t}\right)}^{0.5}}}{{k}^{2}}\right)}^{2} \]

      3. inv-pow 23.8%

         \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {\color{blue}{\left({t}^{-1}\right)}}^{0.5}}{{k}^{2}}\right)}^{2} \]

      4. pow-pow 23.8%

         \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{{t}^{\left(-1 \cdot 0.5\right)}}}{{k}^{2}}\right)}^{2} \]

      5. metadata-eval 23.8%

         \[\leadsto {\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t}^{\color{blue}{-0.5}}}{{k}^{2}}\right)}^{2} \]

   10. Applied egg-rr 23.8%

       \[\leadsto {\color{blue}{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t}^{-0.5}}{{k}^{2}}\right)}}^{2} \]

   if 0.299999999999999989 < t

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in k around 0 66.0%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.3:\\ \;\;\;\;{\left(\frac{\left(\ell \cdot \sqrt{2}\right) \cdot {t}^{-0.5}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\tan k}}{k \cdot {t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+142}:\\ \;\;\;\;\frac{\ell \cdot 2}{{t\_m}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.3e-119)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l)))
    (if (<= k 1.55e+142)
      (* (/ (* l 2.0) (* (pow t_m 3.0) (* (sin k) (tan k)))) (* l 0.5))
      (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.3e-119) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l));
	} else if (k <= 1.55e+142) {
		tmp = ((l * 2.0) / (pow(t_m, 3.0) * (sin(k) * tan(k)))) * (l * 0.5);
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.3e-119) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	} else if (k <= 1.55e+142) {
		tmp = ((l * 2.0) / (Math.pow(t_m, 3.0) * (Math.sin(k) * Math.tan(k)))) * (l * 0.5);
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.3e-119)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	elseif (k <= 1.55e+142)
		tmp = Float64(Float64(Float64(l * 2.0) / Float64((t_m ^ 3.0) * Float64(sin(k) * tan(k)))) * Float64(l * 0.5));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.3e-119], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+142], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.29999999999999993e-119

   1. Initial program 57.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in k around 0 56.2%

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

      1. add-cube-cbrt 59.0%

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

      2. pow3 59.1%

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

      3. cbrt-div 59.0%

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

      4. rem-cbrt-cube 65.3%

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

   7. Applied egg-rr 59.6%

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

   if 2.29999999999999993e-119 < k < 1.55e142

   1. Initial program 66.2%

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

   3. Simplified 66.3%

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

      1. associate-*r* 69.7%

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

      2. *-un-lft-identity 69.7%

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

      3. times-frac 69.5%

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

      4. associate-/l/ 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. /-rgt-identity 69.6%

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

      2. associate-*l/ 69.6%

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

      3. associate-*l* 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in k around 0 72.6%

      \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(0.5 \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative 72.6%

          \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   11. Simplified 72.6%

       \[\leadsto \frac{2 \cdot \ell}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \color{blue}{\left(\ell \cdot 0.5\right)} \]

   if 1.55e142 < k

   1. Initial program 38.5%

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

   3. Simplified 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

   6. Applied egg-rr 26.8%

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

   7. Taylor expanded in t around 0 34.3%

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

   8. Taylor expanded in k around 0 55.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 55.7%

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

      2. unpow2 55.7%

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

      3. rem-square-sqrt 55.7%

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

      4. *-commutative 55.7%

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

   10. Simplified 55.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+142}:\\ \;\;\;\;\frac{\ell \cdot 2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.2e-15)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l)))
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-15) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l));
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-15) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.2e-15)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.2e-15], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.1999999999999999e-15

   1. Initial program 59.5%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in k around 0 59.6%

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

      1. add-cube-cbrt 61.1%

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

      2. pow3 61.1%

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

      3. cbrt-div 61.1%

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

      4. rem-cbrt-cube 66.7%

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

   7. Applied egg-rr 62.7%

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

   if 3.1999999999999999e-15 < k

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in t around 0 38.4%

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

   8. Taylor expanded in k around 0 56.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

      3. rem-square-sqrt 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e+142)
    (/ 2.0 (* (pow (/ t_m (cbrt l)) 3.0) (/ (* 2.0 (pow k 2.0)) l)))
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e+142) {
		tmp = 2.0 / (pow((t_m / cbrt(l)), 3.0) * ((2.0 * pow(k, 2.0)) / l));
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e+142) {
		tmp = 2.0 / (Math.pow((t_m / Math.cbrt(l)), 3.0) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e+142)
		tmp = Float64(2.0 / Float64((Float64(t_m / cbrt(l)) ^ 3.0) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e+142], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.9999999999999998e142

   1. Initial program 59.4%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. associate-*l/ 59.7%

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

   7. Applied egg-rr 59.7%

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

      1. associate-/l* 59.6%

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

   9. Simplified 59.6%

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

       1. add-cube-cbrt 61.6%

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

       2. pow3 61.6%

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

       3. cbrt-div 61.6%

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

       4. rem-cbrt-cube 67.2%

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

   11. Applied egg-rr 62.7%

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

   if 8.9999999999999998e142 < k

   1. Initial program 38.5%

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

   3. Simplified 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

   6. Applied egg-rr 26.8%

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

   7. Taylor expanded in t around 0 34.3%

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

   8. Taylor expanded in k around 0 55.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 55.7%

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

      2. unpow2 55.7%

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

      3. rem-square-sqrt 55.7%

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

      4. *-commutative 55.7%

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

   10. Simplified 55.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 7.2e-207)
    (/ 2.0 (* (* k 2.0) (* (sin k) (/ (pow t_m 3.0) (* l l)))))
    (if (<= k 2.5e-15)
      (/ 2.0 (/ (* (* 2.0 (pow k 2.0)) (/ (pow t_m 3.0) l)) l))
      (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.2e-207) {
		tmp = 2.0 / ((k * 2.0) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	} else if (k <= 2.5e-15) {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.2d-207) then
        tmp = 2.0d0 / ((k * 2.0d0) * (sin(k) * ((t_m ** 3.0d0) / (l * l))))
    else if (k <= 2.5d-15) then
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    else
        tmp = (l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 7.2e-207) {
		tmp = 2.0 / ((k * 2.0) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	} else if (k <= 2.5e-15) {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 7.2e-207:
		tmp = 2.0 / ((k * 2.0) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))
	elif k <= 2.5e-15:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	else:
		tmp = math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 7.2e-207)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	elseif (k <= 2.5e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 7.2e-207)
		tmp = 2.0 / ((k * 2.0) * (sin(k) * ((t_m ^ 3.0) / (l * l))));
	elseif (k <= 2.5e-15)
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	else
		tmp = (l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 7.2e-207], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e-15], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 7.1999999999999993e-207

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in k around 0 54.9%

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

      1. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   if 7.1999999999999993e-207 < k < 2.5e-15

   1. Initial program 67.7%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in k around 0 74.3%

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

      1. associate-*l/ 76.2%

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

   7. Applied egg-rr 76.2%

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

   if 2.5e-15 < k

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in t around 0 38.4%

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

   8. Taylor expanded in k around 0 56.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

      3. rem-square-sqrt 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.3e-15)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (* (pow t_m 2.0) (/ t_m l)) l)))
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.3e-15) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.3d-15) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 2.0d0) * (t_m / l)) / l))
    else
        tmp = (l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.3e-15) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.3e-15:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 2.0) * (t_m / l)) / l))
	else:
		tmp = math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.3e-15)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l)));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.3e-15)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 2.0) * (t_m / l)) / l));
	else
		tmp = (l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.3e-15], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.2999999999999999e-15

   1. Initial program 59.5%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in k around 0 59.6%

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

      1. unpow3 59.6%

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

      2. *-un-lft-identity 59.6%

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

      3. times-frac 62.1%

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

      4. pow2 62.1%

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

   7. Applied egg-rr 62.1%

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

   if 2.2999999999999999e-15 < k

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in t around 0 38.4%

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

   8. Taylor expanded in k around 0 56.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

      3. rem-square-sqrt 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.75e-15)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (/ (pow t_m 3.0) l) l)))
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e-15) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 3.0) / l) / l));
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d-15) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 3.0d0) / l) / l))
    else
        tmp = (l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.75e-15) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 3.0) / l) / l));
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.75e-15:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 3.0) / l) / l))
	else:
		tmp = math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.75e-15)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / l)));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.75e-15)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 3.0) / l) / l));
	else
		tmp = (l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.75e-15], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e-15

   1. Initial program 59.5%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in k around 0 59.6%

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

   if 1.75e-15 < k

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in t around 0 38.4%

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

   8. Taylor expanded in k around 0 56.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

      3. rem-square-sqrt 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.5e+142)
    (/ 2.0 (* (/ (* 2.0 (pow k 2.0)) l) (/ (pow t_m 3.0) l)))
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.5e+142) {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) / l) * (pow(t_m, 3.0) / l));
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.5d+142) then
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) / l) * ((t_m ** 3.0d0) / l))
    else
        tmp = (l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.5e+142) {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) / l) * (Math.pow(t_m, 3.0) / l));
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.5e+142:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) / l) * (math.pow(t_m, 3.0) / l))
	else:
		tmp = math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.5e+142)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64((t_m ^ 3.0) / l)));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.5e+142)
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) / l) * ((t_m ^ 3.0) / l));
	else
		tmp = (l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.5e+142], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.50000000000000035e142

   1. Initial program 59.4%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. associate-*l/ 59.7%

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

   7. Applied egg-rr 59.7%

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

      1. associate-/l* 59.6%

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

   9. Simplified 59.6%

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

   if 5.50000000000000035e142 < k

   1. Initial program 38.5%

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

   3. Simplified 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. pow2 38.5%

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

   6. Applied egg-rr 26.8%

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

   7. Taylor expanded in t around 0 34.3%

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

   8. Taylor expanded in k around 0 55.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 55.7%

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

      2. unpow2 55.7%

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

      3. rem-square-sqrt 55.7%

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

      4. *-commutative 55.7%

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

   10. Simplified 55.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.2e-15)
    (/ 2.0 (/ (* (* 2.0 (pow k 2.0)) (/ (pow t_m 3.0) l)) l))
    (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-15) {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) * (pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.2d-15) then
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) * ((t_m ** 3.0d0) / l)) / l)
    else
        tmp = (l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-15) {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) * (Math.pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.2e-15:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) * (math.pow(t_m, 3.0) / l)) / l)
	else:
		tmp = math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.2e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) * Float64((t_m ^ 3.0) / l)) / l));
	else
		tmp = Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.2e-15)
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) * ((t_m ^ 3.0) / l)) / l);
	else
		tmp = (l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.2e-15], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.1999999999999999e-15

   1. Initial program 59.5%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in k around 0 59.6%

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

      1. associate-*l/ 60.6%

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

   7. Applied egg-rr 60.6%

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

   if 3.1999999999999999e-15 < k

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

   6. Applied egg-rr 36.6%

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

   7. Taylor expanded in t around 0 38.4%

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

   8. Taylor expanded in k around 0 56.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 56.8%

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

      2. unpow2 56.8%

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

      3. rem-square-sqrt 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 51.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left({\ell}^{2} \cdot \frac{2}{t\_m \cdot {k}^{4}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (pow l 2.0) (/ 2.0 (* t_m (pow k 4.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (pow(l, 2.0) * (2.0 / (t_m * pow(k, 4.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l ** 2.0d0) * (2.0d0 / (t_m * (k ** 4.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (Math.pow(l, 2.0) * (2.0 / (t_m * Math.pow(k, 4.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (math.pow(l, 2.0) * (2.0 / (t_m * math.pow(k, 4.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64((l ^ 2.0) * Float64(2.0 / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l ^ 2.0) * (2.0 / (t_m * (k ^ 4.0))));
end

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 57.6%

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

   1. add-sqr-sqrt 44.7%

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

   2. pow2 44.7%

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

6. Applied egg-rr 45.0%

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

7. Taylor expanded in t around 0 38.2%

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

8. Taylor expanded in k around 0 54.7%

   \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
9. Step-by-step derivation

   1. associate-/l* 54.7%

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

   2. unpow2 54.7%

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

   3. rem-square-sqrt 54.7%

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

   4. *-commutative 54.7%

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

10. Simplified 54.7%

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

11. Final simplification 54.7%

    \[\leadsto {\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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))))