Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 82.2%

Time: 31.3s

Alternatives: 24

Speedup: 1.9×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% 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: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \tan k\_m \cdot \sin k\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.3 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k\_m}}{t\_2} \cdot t\_m\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}\right)\right)\right)}\\ \mathbf{elif}\;k\_m \leq 1.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right) \cdot \sqrt{t\_3}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{t\_2} \cdot \sqrt[3]{t\_3 \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0)) (t_3 (* (tan k_m) (sin k_m))))
   (*
    t_s
    (if (<= k_m 1.3e-115)
      (/
       2.0
       (*
        (pow (* (/ (cbrt k_m) t_2) t_m) 3.0)
        (* (tan k_m) (+ 1.0 (+ 1.0 (/ (/ k_m t_m) (/ t_m k_m)))))))
      (if (<= k_m 1.6)
        (/
         2.0
         (pow
          (*
           (/ (pow t_m 1.5) l)
           (* (hypot 1.0 (hypot 1.0 (/ k_m t_m))) (sqrt t_3)))
          2.0))
        (if (<= k_m 1.2e+154)
          (/
           2.0
           (/
            (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
            (* (pow l 2.0) (cos k_m))))
          (/
           2.0
           (pow
            (* (/ t_m t_2) (cbrt (* t_3 (+ 2.0 (pow (/ k_m t_m) 2.0)))))
            3.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), 2.0);
	double t_3 = tan(k_m) * sin(k_m);
	double tmp;
	if (k_m <= 1.3e-115) {
		tmp = 2.0 / (pow(((cbrt(k_m) / t_2) * t_m), 3.0) * (tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))));
	} else if (k_m <= 1.6) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t_m))) * sqrt(t_3))), 2.0);
	} else if (k_m <= 1.2e+154) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	} else {
		tmp = 2.0 / pow(((t_m / t_2) * cbrt((t_3 * (2.0 + pow((k_m / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.tan(k_m) * Math.sin(k_m);
	double tmp;
	if (k_m <= 1.3e-115) {
		tmp = 2.0 / (Math.pow(((Math.cbrt(k_m) / t_2) * t_m), 3.0) * (Math.tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))));
	} else if (k_m <= 1.6) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))) * Math.sqrt(t_3))), 2.0);
	} else if (k_m <= 1.2e+154) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / t_2) * Math.cbrt((t_3 * (2.0 + Math.pow((k_m / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ 2.0
	t_3 = Float64(tan(k_m) * sin(k_m))
	tmp = 0.0
	if (k_m <= 1.3e-115)
		tmp = Float64(2.0 / Float64((Float64(Float64(cbrt(k_m) / t_2) * t_m) ^ 3.0) * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m)))))));
	elseif (k_m <= 1.6)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t_m))) * sqrt(t_3))) ^ 2.0));
	elseif (k_m <= 1.2e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / t_2) * cbrt(Float64(t_3 * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.3e-115], N[(2.0 / N[(N[Power[N[(N[(N[Power[k$95$m, 1/3], $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$m), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.6], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.2e+154], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[Power[N[(t$95$3 * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.30000000000000002e-115

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

      3. *-commutative 58.5%

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

      4. cbrt-prod 58.4%

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

      5. cbrt-div 59.0%

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

      6. rem-cbrt-cube 65.0%

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

      7. cbrt-prod 80.0%

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

      8. pow2 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. *-commutative 80.0%

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

   8. Simplified 80.0%

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

      1. unpow2 80.0%

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

      2. clear-num 80.0%

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

      3. un-div-inv 80.0%

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

   10. Applied egg-rr 80.0%

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

   11. Taylor expanded in k around 0 60.7%

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

       1. cbrt-div 62.3%

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

       2. pow2 62.3%

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

       3. cbrt-prod 76.3%

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

       4. unpow2 76.3%

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

   13. Applied egg-rr 76.3%

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

   if 1.30000000000000002e-115 < k < 1.6000000000000001

   1. Initial program 65.6%

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

   4. Applied egg-rr 57.2%

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

   if 1.6000000000000001 < k < 1.20000000000000007e154

   1. Initial program 68.2%

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

   3. Taylor expanded in t around 0 86.2%

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

      1. associate-*r* 86.2%

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

   5. Simplified 86.2%

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

      1. unpow2 86.2%

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

      2. sin-mult 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. div-sub 86.2%

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

      2. +-inverses 86.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 86.2%

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

      4. metadata-eval 86.2%

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

      5. count-2 86.2%

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

      6. *-commutative 86.2%

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

   9. Simplified 86.2%

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

   if 1.20000000000000007e154 < k

   1. Initial program 34.9%

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

      1. add-cube-cbrt 34.9%

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

      2. pow3 34.9%

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

   4. Applied egg-rr 73.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\tan k \cdot \sin k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(\tan k \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k\_m}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\_m\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}\right)\right)\right)}\\ \mathbf{elif}\;k\_m \leq 1.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right) \cdot \sqrt{\tan k\_m \cdot \sin k\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.9e-112)
    (/
     2.0
     (*
      (pow (* (/ (cbrt k_m) (pow (cbrt l) 2.0)) t_m) 3.0)
      (* (tan k_m) (+ 1.0 (+ 1.0 (/ (/ k_m t_m) (/ t_m k_m)))))))
    (if (<= k_m 1.6)
      (/
       2.0
       (pow
        (*
         (/ (pow t_m 1.5) l)
         (*
          (hypot 1.0 (hypot 1.0 (/ k_m t_m)))
          (sqrt (* (tan k_m) (sin k_m)))))
        2.0))
      (if (<= k_m 1.35e+154)
        (/
         2.0
         (/
          (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
          (* (pow l 2.0) (cos k_m))))
        (/
         2.0
         (pow
          (*
           (* t_m (pow (cbrt l) -2.0))
           (cbrt (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0))))))
          3.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-112) {
		tmp = 2.0 / (pow(((cbrt(k_m) / pow(cbrt(l), 2.0)) * t_m), 3.0) * (tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))));
	} else if (k_m <= 1.6) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t_m))) * sqrt((tan(k_m) * sin(k_m))))), 2.0);
	} else if (k_m <= 1.35e+154) {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	} else {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 1.9e-112) {
		tmp = 2.0 / (Math.pow(((Math.cbrt(k_m) / Math.pow(Math.cbrt(l), 2.0)) * t_m), 3.0) * (Math.tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))));
	} else if (k_m <= 1.6) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))) * Math.sqrt((Math.tan(k_m) * Math.sin(k_m))))), 2.0);
	} else if (k_m <= 1.35e+154) {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	} else {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-112)
		tmp = Float64(2.0 / Float64((Float64(Float64(cbrt(k_m) / (cbrt(l) ^ 2.0)) * t_m) ^ 3.0) * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m)))))));
	elseif (k_m <= 1.6)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t_m))) * sqrt(Float64(tan(k_m) * sin(k_m))))) ^ 2.0));
	elseif (k_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-112], N[(2.0 / N[(N[Power[N[(N[(N[Power[k$95$m, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.6], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.35e+154], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 1.89999999999999997e-112

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

      3. *-commutative 58.5%

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

      4. cbrt-prod 58.4%

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

      5. cbrt-div 59.0%

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

      6. rem-cbrt-cube 65.0%

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

      7. cbrt-prod 80.0%

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

      8. pow2 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. *-commutative 80.0%

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

   8. Simplified 80.0%

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

      1. unpow2 80.0%

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

      2. clear-num 80.0%

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

      3. un-div-inv 80.0%

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

   10. Applied egg-rr 80.0%

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

   11. Taylor expanded in k around 0 60.7%

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

       1. cbrt-div 62.3%

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

       2. pow2 62.3%

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

       3. cbrt-prod 76.3%

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

       4. unpow2 76.3%

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

   13. Applied egg-rr 76.3%

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

   if 1.89999999999999997e-112 < k < 1.6000000000000001

   1. Initial program 65.6%

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

   4. Applied egg-rr 57.2%

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

   if 1.6000000000000001 < k < 1.35000000000000003e154

   1. Initial program 68.2%

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

   3. Taylor expanded in t around 0 86.2%

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

      1. associate-*r* 86.2%

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

   5. Simplified 86.2%

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

      1. unpow2 86.2%

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

      2. sin-mult 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. div-sub 86.2%

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

      2. +-inverses 86.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 86.2%

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

      4. metadata-eval 86.2%

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

      5. count-2 86.2%

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

      6. *-commutative 86.2%

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

   9. Simplified 86.2%

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

   if 1.35000000000000003e154 < k

   1. Initial program 34.9%

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

   3. Simplified 35.0%

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

      1. add-cube-cbrt 34.9%

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

      2. pow3 34.9%

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

      3. *-commutative 34.9%

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

      4. cbrt-prod 34.8%

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

      5. cbrt-div 34.9%

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

      6. rem-cbrt-cube 46.6%

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

      7. cbrt-prod 54.0%

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

      8. pow2 54.0%

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

   6. Applied egg-rr 54.0%

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

      1. *-commutative 54.0%

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

   8. Simplified 54.0%

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

   10. Applied egg-rr 73.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\tan k \cdot \sin k}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k\_m}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\_m\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}\right)\right)\right)}\\ \mathbf{elif}\;k\_m \leq 1.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right) \cdot \sqrt{\tan k\_m \cdot \sin k\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2e-115)
    (/
     2.0
     (*
      (pow (* (/ (cbrt k_m) (pow (cbrt l) 2.0)) t_m) 3.0)
      (* (tan k_m) (+ 1.0 (+ 1.0 (/ (/ k_m t_m) (/ t_m k_m)))))))
    (if (<= k_m 1.6)
      (/
       2.0
       (pow
        (*
         (/ (pow t_m 1.5) l)
         (*
          (hypot 1.0 (hypot 1.0 (/ k_m t_m)))
          (sqrt (* (tan k_m) (sin k_m)))))
        2.0))
      (/
       2.0
       (/
        (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
        (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2e-115) {
		tmp = 2.0 / (pow(((cbrt(k_m) / pow(cbrt(l), 2.0)) * t_m), 3.0) * (tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))));
	} else if (k_m <= 1.6) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k_m / t_m))) * sqrt((tan(k_m) * sin(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2e-115) {
		tmp = 2.0 / (Math.pow(((Math.cbrt(k_m) / Math.pow(Math.cbrt(l), 2.0)) * t_m), 3.0) * (Math.tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))));
	} else if (k_m <= 1.6) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))) * Math.sqrt((Math.tan(k_m) * Math.sin(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2e-115)
		tmp = Float64(2.0 / Float64((Float64(Float64(cbrt(k_m) / (cbrt(l) ^ 2.0)) * t_m) ^ 3.0) * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m)))))));
	elseif (k_m <= 1.6)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k_m / t_m))) * sqrt(Float64(tan(k_m) * sin(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2e-115], N[(2.0 / N[(N[Power[N[(N[(N[Power[k$95$m, 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.6], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.0000000000000001e-115

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

      3. *-commutative 58.5%

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

      4. cbrt-prod 58.4%

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

      5. cbrt-div 59.0%

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

      6. rem-cbrt-cube 65.0%

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

      7. cbrt-prod 80.0%

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

      8. pow2 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. *-commutative 80.0%

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

   8. Simplified 80.0%

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

      1. unpow2 80.0%

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

      2. clear-num 80.0%

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

      3. un-div-inv 80.0%

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

   10. Applied egg-rr 80.0%

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

   11. Taylor expanded in k around 0 60.7%

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

       1. cbrt-div 62.3%

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

       2. pow2 62.3%

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

       3. cbrt-prod 76.3%

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

       4. unpow2 76.3%

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

   13. Applied egg-rr 76.3%

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

   if 2.0000000000000001e-115 < k < 1.6000000000000001

   1. Initial program 65.6%

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

   4. Applied egg-rr 57.2%

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

   if 1.6000000000000001 < k

   1. Initial program 52.2%

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

   3. Taylor expanded in t around 0 71.0%

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

      1. associate-*r* 70.9%

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

   5. Simplified 70.9%

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

      1. unpow2 70.9%

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

      2. sin-mult 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. div-sub 71.0%

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

      2. +-inverses 71.0%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 71.0%

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

      4. metadata-eval 71.0%

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

      5. count-2 71.0%

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

      6. *-commutative 71.0%

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

   9. Simplified 71.0%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.6:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\tan k \cdot \sin k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 110000000000:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m \cdot \sqrt[3]{\sin k\_m}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{2 \cdot \sin k\_m}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 110000000000.0)
    (/
     2.0
     (*
      (pow (/ (* t_m (cbrt (sin k_m))) (pow (cbrt l) 2.0)) 3.0)
      (/ (* 2.0 (sin k_m)) (cos k_m))))
    (/
     2.0
     (/
      (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
      (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 110000000000.0) {
		tmp = 2.0 / (pow(((t_m * cbrt(sin(k_m))) / pow(cbrt(l), 2.0)), 3.0) * ((2.0 * sin(k_m)) / cos(k_m)));
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 110000000000.0) {
		tmp = 2.0 / (Math.pow(((t_m * Math.cbrt(Math.sin(k_m))) / Math.pow(Math.cbrt(l), 2.0)), 3.0) * ((2.0 * Math.sin(k_m)) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 110000000000.0)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m * cbrt(sin(k_m))) / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(Float64(2.0 * sin(k_m)) / cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 110000000000.0], N[(2.0 / N[(N[Power[N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(2.0 * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1e11

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. add-cube-cbrt 59.1%

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

      2. pow3 59.1%

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

      3. *-commutative 59.1%

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

      4. cbrt-prod 59.0%

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

      5. cbrt-div 59.5%

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

      6. rem-cbrt-cube 65.2%

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

      7. cbrt-prod 79.6%

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

      8. pow2 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

   8. Simplified 79.6%

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

      1. unpow2 79.6%

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

      2. clear-num 79.6%

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

      3. un-div-inv 79.6%

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

   10. Applied egg-rr 79.6%

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

       1. associate-*l/ 79.6%

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

   12. Applied egg-rr 79.6%

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

   13. Taylor expanded in t around inf 77.1%

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

       1. *-commutative 77.1%

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

       2. associate-*l/ 77.1%

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

   15. Simplified 77.1%

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

   if 1.1e11 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 110000000000:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{2 \cdot \sin k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{t\_m \cdot \sqrt[3]{\sin k\_m}}{t\_2}\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 180000000000:\\ \;\;\;\;\frac{2}{\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\frac{t\_m}{t\_2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) 2.0)))
   (*
    t_s
    (if (<= k_m 2.5e-14)
      (/ 2.0 (* (* k_m 2.0) (pow (/ (* t_m (cbrt (sin k_m))) t_2) 3.0)))
      (if (<= k_m 180000000000.0)
        (/
         2.0
         (*
          (* (* (tan k_m) (sin k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0)))
          (pow (/ t_m t_2) 3.0)))
        (/
         2.0
         (/
          (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
          (* (pow l 2.0) (cos k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 2.5e-14) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((t_m * cbrt(sin(k_m))) / t_2), 3.0));
	} else if (k_m <= 180000000000.0) {
		tmp = 2.0 / (((tan(k_m) * sin(k_m)) * (2.0 + pow((k_m / t_m), 2.0))) * pow((t_m / t_2), 3.0));
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 2.5e-14) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((t_m * Math.cbrt(Math.sin(k_m))) / t_2), 3.0));
	} else if (k_m <= 180000000000.0) {
		tmp = 2.0 / (((Math.tan(k_m) * Math.sin(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0))) * Math.pow((t_m / t_2), 3.0));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 2.5e-14)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64(t_m * cbrt(sin(k_m))) / t_2) ^ 3.0)));
	elseif (k_m <= 180000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))) * (Float64(t_m / t_2) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.5e-14], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 180000000000.0], N[(2.0 / N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m / t$95$2), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.5000000000000001e-14

   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 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.3%

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

      5. cbrt-div 59.8%

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

      6. rem-cbrt-cube 65.2%

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

      7. cbrt-prod 79.3%

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

      8. pow2 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. *-commutative 79.3%

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

   8. Simplified 79.3%

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

      1. unpow2 79.3%

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

      2. clear-num 79.3%

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

      3. un-div-inv 79.3%

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

   10. Applied egg-rr 79.3%

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

       1. associate-*l/ 79.3%

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

   12. Applied egg-rr 79.3%

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

   13. Taylor expanded in k around 0 74.9%

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

       1. *-commutative 63.9%

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

   15. Simplified 74.9%

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

   if 2.5000000000000001e-14 < k < 1.8e11

   1. Initial program 35.9%

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

      1. add-cube-cbrt 35.9%

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

      2. pow3 35.9%

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

   4. Applied egg-rr 98.1%

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

      1. cube-prod 98.4%

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

      2. rem-cube-cbrt 99.0%

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

   6. Simplified 99.0%

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

   if 1.8e11 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 180000000000:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 66000000000:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(1 + \left(1 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}\right)\right)\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 66000000000.0)
    (/
     2.0
     (*
      (* (tan k_m) (+ 1.0 (+ 1.0 (/ (/ k_m t_m) (/ t_m k_m)))))
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0)))
    (/
     2.0
     (/
      (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
      (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 66000000000.0) {
		tmp = 2.0 / ((tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))) * pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0));
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 66000000000.0) {
		tmp = 2.0 / ((Math.tan(k_m) * (1.0 + (1.0 + ((k_m / t_m) / (t_m / k_m))))) * Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 66000000000.0)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m))))) * (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 66000000000.0], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.6e10

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. add-cube-cbrt 59.1%

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

      2. pow3 59.1%

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

      3. *-commutative 59.1%

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

      4. cbrt-prod 59.0%

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

      5. cbrt-div 59.5%

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

      6. rem-cbrt-cube 65.2%

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

      7. cbrt-prod 79.6%

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

      8. pow2 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

   8. Simplified 79.6%

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

      1. unpow2 79.6%

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

      2. clear-num 79.6%

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

      3. un-div-inv 79.6%

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

   10. Applied egg-rr 79.6%

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

   if 6.6e10 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 66000000000:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{t\_m \cdot \sqrt[3]{\sin k\_m}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 195000000000:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \left(2 + \frac{k\_m}{t\_m \cdot \frac{t\_m}{k\_m}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 8.5e-12)
    (/
     2.0
     (* (* k_m 2.0) (pow (/ (* t_m (cbrt (sin k_m))) (pow (cbrt l) 2.0)) 3.0)))
    (if (<= k_m 195000000000.0)
      (/
       2.0
       (*
        (pow (* t_m (pow (cbrt l) -2.0)) 3.0)
        (* (* (tan k_m) (sin k_m)) (+ 2.0 (/ k_m (* t_m (/ t_m k_m)))))))
      (/
       2.0
       (/
        (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
        (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 8.5e-12) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((t_m * cbrt(sin(k_m))) / pow(cbrt(l), 2.0)), 3.0));
	} else if (k_m <= 195000000000.0) {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * ((tan(k_m) * sin(k_m)) * (2.0 + (k_m / (t_m * (t_m / k_m))))));
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 8.5e-12) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((t_m * Math.cbrt(Math.sin(k_m))) / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else if (k_m <= 195000000000.0) {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * ((Math.tan(k_m) * Math.sin(k_m)) * (2.0 + (k_m / (t_m * (t_m / k_m))))));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 8.5e-12)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64(t_m * cbrt(sin(k_m))) / (cbrt(l) ^ 2.0)) ^ 3.0)));
	elseif (k_m <= 195000000000.0)
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(Float64(tan(k_m) * sin(k_m)) * Float64(2.0 + Float64(k_m / Float64(t_m * Float64(t_m / k_m)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 8.5e-12], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 195000000000.0], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 8.4999999999999997e-12

   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 59.5%

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

      1. add-cube-cbrt 59.4%

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

      2. pow3 59.4%

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

      3. *-commutative 59.4%

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

      4. cbrt-prod 59.3%

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

      5. cbrt-div 59.8%

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

      6. rem-cbrt-cube 65.2%

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

      7. cbrt-prod 79.3%

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

      8. pow2 79.3%

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

   6. Applied egg-rr 79.3%

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

      1. *-commutative 79.3%

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

   8. Simplified 79.3%

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

      1. unpow2 79.3%

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

      2. clear-num 79.3%

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

      3. un-div-inv 79.3%

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

   10. Applied egg-rr 79.3%

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

       1. associate-*l/ 79.3%

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

   12. Applied egg-rr 79.3%

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

   13. Taylor expanded in k around 0 74.9%

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

       1. *-commutative 63.9%

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

   15. Simplified 74.9%

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

   if 8.4999999999999997e-12 < k < 1.95e11

   1. Initial program 35.9%

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

   3. Simplified 67.9%

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

      1. unpow2 67.9%

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

      2. clear-num 67.9%

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

      3. frac-times 67.9%

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

      4. *-un-lft-identity 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. add-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

      3. associate-/l/ 35.9%

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

      4. cbrt-div 35.9%

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

      5. unpow3 35.9%

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

      6. add-cbrt-cube 66.9%

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

      7. cbrt-prod 98.4%

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

      8. unpow2 98.4%

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

      9. div-inv 98.4%

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

      10. unpow-prod-down 35.9%

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

      11. pow-flip 35.9%

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

      12. metadata-eval 35.9%

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

   8. Applied egg-rr 35.9%

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

      1. cube-prod 97.5%

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

   10. Simplified 97.5%

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

   if 1.95e11 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 195000000000:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 31000000000:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot {k\_m}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.9e-151)
    (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k_m))) 2.0)))
    (if (<= k_m 31000000000.0)
      (/
       2.0
       (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (* 2.0 (pow k_m 2.0)))) 3.0))
      (/
       2.0
       (/
        (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
        (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-151) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else if (k_m <= 31000000000.0) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((2.0 * pow(k_m, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 1.9e-151) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else if (k_m <= 31000000000.0) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((2.0 * Math.pow(k_m, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-151)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	elseif (k_m <= 31000000000.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(2.0 * (k_m ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-151], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 31000000000.0], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.89999999999999985e-151

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

      3. *-commutative 28.9%

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

      4. sqrt-prod 12.2%

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

      5. sqrt-div 12.2%

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

      6. sqrt-pow1 13.4%

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

      7. metadata-eval 13.4%

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

      8. sqrt-prod 7.3%

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

      9. add-sqr-sqrt 14.6%

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

   6. Applied egg-rr 14.6%

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

      1. *-commutative 14.6%

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

   8. Simplified 14.6%

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

   9. Taylor expanded in k around 0 13.6%

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

   if 1.89999999999999985e-151 < k < 3.1e10

   1. Initial program 64.6%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in k around 0 78.3%

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

      1. add-cube-cbrt 78.2%

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

      2. pow3 78.3%

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

      3. cbrt-prod 78.0%

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

      4. associate-/l/ 75.0%

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

      5. cbrt-div 75.1%

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

      6. unpow3 75.1%

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

      7. add-cbrt-cube 77.7%

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

      8. cbrt-prod 84.9%

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

      9. unpow2 84.9%

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

      10. div-inv 84.9%

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

      11. pow-flip 84.9%

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

      12. metadata-eval 84.9%

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

   7. Applied egg-rr 84.9%

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

   if 3.1e10 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 31000000000:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 35000000000:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{t\_m \cdot \sqrt[3]{\sin k\_m}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 35000000000.0)
    (/
     2.0
     (* (* k_m 2.0) (pow (/ (* t_m (cbrt (sin k_m))) (pow (cbrt l) 2.0)) 3.0)))
    (/
     2.0
     (/
      (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
      (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 35000000000.0) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((t_m * cbrt(sin(k_m))) / pow(cbrt(l), 2.0)), 3.0));
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 35000000000.0) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((t_m * Math.cbrt(Math.sin(k_m))) / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 35000000000.0)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64(t_m * cbrt(sin(k_m))) / (cbrt(l) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 35000000000.0], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.5e10

   1. Initial program 59.2%

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

   3. Simplified 59.2%

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

      1. add-cube-cbrt 59.1%

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

      2. pow3 59.1%

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

      3. *-commutative 59.1%

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

      4. cbrt-prod 59.0%

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

      5. cbrt-div 59.5%

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

      6. rem-cbrt-cube 65.2%

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

      7. cbrt-prod 79.6%

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

      8. pow2 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. *-commutative 79.6%

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

   8. Simplified 79.6%

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

      1. unpow2 79.6%

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

      2. clear-num 79.6%

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

      3. un-div-inv 79.6%

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

   10. Applied egg-rr 79.6%

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

       1. associate-*l/ 79.6%

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

   12. Applied egg-rr 79.6%

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

   13. Taylor expanded in k around 0 74.3%

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

       1. *-commutative 63.5%

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

   15. Simplified 74.3%

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

   if 3.5e10 < k

   1. Initial program 53.1%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.2%

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

   5. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. sin-mult 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. div-sub 72.2%

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

      2. +-inverses 72.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 72.2%

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

      4. metadata-eval 72.2%

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

      5. count-2 72.2%

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

      6. *-commutative 72.2%

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

   9. Simplified 72.2%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 35000000000:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 9.5 \cdot 10^{+93}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m}\right) \cdot \frac{\ell}{2 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {k\_m}^{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.35e-5)
    (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k_m))) 2.0)))
    (if (<= k_m 9.5e+93)
      (*
       (* (/ (/ 2.0 (pow t_m 3.0)) (sin k_m)) (/ l (tan k_m)))
       (/ l (+ 2.0 (/ (/ k_m t_m) (/ t_m k_m)))))
      (/
       2.0
       (/
        (* (pow k_m 2.0) (* t_m (pow k_m 2.0)))
        (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.35e-5) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else if (k_m <= 9.5e+93) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k_m)) * (l / tan(k_m))) * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(k_m, 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.35d-5) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k_m))) ** 2.0d0))
    else if (k_m <= 9.5d+93) then
        tmp = (((2.0d0 / (t_m ** 3.0d0)) / sin(k_m)) * (l / tan(k_m))) * (l / (2.0d0 + ((k_m / t_m) / (t_m / k_m))))
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * (t_m * (k_m ** 2.0d0))) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2.35e-5) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else if (k_m <= 9.5e+93) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k_m)) * (l / Math.tan(k_m))) * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(k_m, 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.35e-5:
		tmp = 2.0 / ((k_m * 2.0) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k_m))), 2.0))
	elif k_m <= 9.5e+93:
		tmp = (((2.0 / math.pow(t_m, 3.0)) / math.sin(k_m)) * (l / math.tan(k_m))) * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))))
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * (t_m * math.pow(k_m, 2.0))) / (math.pow(l, 2.0) * math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.35e-5)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	elseif (k_m <= 9.5e+93)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k_m)) * Float64(l / tan(k_m))) * Float64(l / Float64(2.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (k_m ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.35e-5)
		tmp = 2.0 / ((k_m * 2.0) * ((((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0));
	elseif (k_m <= 9.5e+93)
		tmp = (((2.0 / (t_m ^ 3.0)) / sin(k_m)) * (l / tan(k_m))) * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	else
		tmp = 2.0 / (((k_m ^ 2.0) * (t_m * (k_m ^ 2.0))) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.35e-5], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 9.5e+93], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.34999999999999986e-5

   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 59.5%

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

      1. add-sqr-sqrt 30.0%

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

      2. pow2 30.0%

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

      3. *-commutative 30.0%

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

      4. sqrt-prod 16.4%

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

      5. sqrt-div 16.4%

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

      6. sqrt-pow1 17.9%

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

      7. metadata-eval 17.9%

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

      8. sqrt-prod 10.8%

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

      9. add-sqr-sqrt 21.2%

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

   6. Applied egg-rr 21.2%

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

      1. *-commutative 21.2%

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

   8. Simplified 21.2%

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

   9. Taylor expanded in k around 0 19.1%

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

   if 2.34999999999999986e-5 < k < 9.4999999999999991e93

   1. Initial program 75.6%

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

   3. Simplified 75.7%

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

      1. associate-*r* 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. /-rgt-identity 87.4%

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

      2. associate-*l/ 87.4%

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

      3. associate-*r* 87.4%

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

      4. times-frac 87.4%

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

      5. associate-/r* 87.3%

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

   8. Simplified 87.3%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

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

   10. Applied egg-rr 87.3%

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

   if 9.4999999999999991e93 < k

   1. Initial program 42.3%

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

   3. Taylor expanded in t around 0 66.2%

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

      1. associate-*r* 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in k around 0 58.7%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+93}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.35:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k\_m}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.35)
    (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k_m))) 2.0)))
    (/
     2.0
     (/
      (* (* t_m (pow k_m 2.0)) (- 0.5 (/ (cos (* k_m 2.0)) 2.0)))
      (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.35) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / (((t_m * pow(k_m, 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.35d0) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k_m))) ** 2.0d0))
    else
        tmp = 2.0d0 / (((t_m * (k_m ** 2.0d0)) * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0))) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 1.35) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k_m, 2.0)) * (0.5 - (Math.cos((k_m * 2.0)) / 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.35:
		tmp = 2.0 / ((k_m * 2.0) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k_m))), 2.0))
	else:
		tmp = 2.0 / (((t_m * math.pow(k_m, 2.0)) * (0.5 - (math.cos((k_m * 2.0)) / 2.0))) / (math.pow(l, 2.0) * math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.35)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k_m ^ 2.0)) * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.35)
		tmp = 2.0 / ((k_m * 2.0) * ((((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0));
	else
		tmp = 2.0 / (((t_m * (k_m ^ 2.0)) * (0.5 - (cos((k_m * 2.0)) / 2.0))) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.35], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.3500000000000001

   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 59.5%

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

      1. add-sqr-sqrt 30.0%

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

      2. pow2 30.0%

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

      3. *-commutative 30.0%

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

      4. sqrt-prod 16.4%

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

      5. sqrt-div 16.4%

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

      6. sqrt-pow1 17.9%

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

      7. metadata-eval 17.9%

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

      8. sqrt-prod 10.8%

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

      9. add-sqr-sqrt 21.2%

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

   6. Applied egg-rr 21.2%

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

      1. *-commutative 21.2%

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

   8. Simplified 21.2%

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

   9. Taylor expanded in k around 0 19.1%

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

   if 1.3500000000000001 < k

   1. Initial program 52.2%

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

   3. Taylor expanded in t around 0 71.0%

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

      1. associate-*r* 70.9%

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

   5. Simplified 70.9%

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

      1. unpow2 70.9%

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

      2. sin-mult 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. div-sub 71.0%

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

      2. +-inverses 71.0%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      3. cos-0 71.0%

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

      4. metadata-eval 71.0%

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

      5. count-2 71.0%

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

      6. *-commutative 71.0%

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

   9. Simplified 71.0%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 3.15 \cdot 10^{+94}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m}\right) \cdot \frac{\ell}{2 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k\_m}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.9e-5)
    (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k_m))) 2.0)))
    (if (<= k_m 3.15e+94)
      (*
       (* (/ (/ 2.0 (pow t_m 3.0)) (sin k_m)) (/ l (tan k_m)))
       (/ l (+ 2.0 (/ (/ k_m t_m) (/ t_m k_m)))))
      (/ 2.0 (* (* k_m 2.0) (pow (* t_m (cbrt (/ k_m (pow l 2.0)))) 3.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-5) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else if (k_m <= 3.15e+94) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k_m)) * (l / tan(k_m))) * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * pow((t_m * cbrt((k_m / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 1.9e-5) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else if (k_m <= 3.15e+94) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k_m)) * (l / Math.tan(k_m))) * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow((t_m * Math.cbrt((k_m / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-5)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	elseif (k_m <= 3.15e+94)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k_m)) * Float64(l / tan(k_m))) * Float64(l / Float64(2.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(t_m * cbrt(Float64(k_m / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.9e-5], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.15e+94], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9000000000000001e-5

   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 59.5%

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

      1. add-sqr-sqrt 30.0%

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

      2. pow2 30.0%

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

      3. *-commutative 30.0%

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

      4. sqrt-prod 16.4%

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

      5. sqrt-div 16.4%

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

      6. sqrt-pow1 17.9%

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

      7. metadata-eval 17.9%

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

      8. sqrt-prod 10.8%

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

      9. add-sqr-sqrt 21.2%

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

   6. Applied egg-rr 21.2%

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

      1. *-commutative 21.2%

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

   8. Simplified 21.2%

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

   9. Taylor expanded in k around 0 19.1%

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

   if 1.9000000000000001e-5 < k < 3.15e94

   1. Initial program 75.6%

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

   3. Simplified 75.7%

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

      1. associate-*r* 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. /-rgt-identity 87.4%

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

      2. associate-*l/ 87.4%

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

      3. associate-*r* 87.4%

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

      4. times-frac 87.4%

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

      5. associate-/r* 87.3%

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

   8. Simplified 87.3%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

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

   10. Applied egg-rr 87.3%

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

   if 3.15e94 < k

   1. Initial program 42.3%

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

   3. Simplified 42.4%

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

      1. add-cube-cbrt 42.3%

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

      2. pow3 42.3%

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

      3. *-commutative 42.3%

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

      4. cbrt-prod 42.3%

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

      5. cbrt-div 42.3%

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

      6. rem-cbrt-cube 53.0%

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

      7. cbrt-prod 58.0%

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

      8. pow2 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. *-commutative 58.0%

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

   8. Simplified 58.0%

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

      1. unpow2 58.0%

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

      2. clear-num 58.0%

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

      3. un-div-inv 58.0%

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

   10. Applied egg-rr 58.0%

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

   11. Taylor expanded in k around 0 58.6%

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

   12. Taylor expanded in k around 0 56.2%

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

       1. *-commutative 56.2%

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

   14. Simplified 56.2%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.15 \cdot 10^{+94}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{2}{{t\_m}^{3}}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;t\_2 \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;k\_m \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{1.5} \cdot \frac{{t\_m}^{1.5}}{\ell}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;t\_2 \cdot \frac{\ell}{2 + \frac{\frac{k\_m}{t\_m}}{\frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k\_m}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (/ (/ 2.0 (pow t_m 3.0)) (sin k_m)) (/ l (tan k_m)))))
   (*
    t_s
    (if (<= k_m 3.3e-167)
      (* t_2 (* l 0.5))
      (if (<= k_m 4.8e-6)
        (/
         2.0
         (* (* 2.0 (pow k_m 2.0)) (/ (* (pow t_m 1.5) (/ (pow t_m 1.5) l)) l)))
        (if (<= k_m 1.9e+94)
          (* t_2 (/ l (+ 2.0 (/ (/ k_m t_m) (/ t_m k_m)))))
          (/
           2.0
           (* (* k_m 2.0) (pow (* t_m (cbrt (/ k_m (pow l 2.0)))) 3.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = ((2.0 / pow(t_m, 3.0)) / sin(k_m)) * (l / tan(k_m));
	double tmp;
	if (k_m <= 3.3e-167) {
		tmp = t_2 * (l * 0.5);
	} else if (k_m <= 4.8e-6) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t_m, 1.5) * (pow(t_m, 1.5) / l)) / l));
	} else if (k_m <= 1.9e+94) {
		tmp = t_2 * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * pow((t_m * cbrt((k_m / pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = ((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k_m)) * (l / Math.tan(k_m));
	double tmp;
	if (k_m <= 3.3e-167) {
		tmp = t_2 * (l * 0.5);
	} else if (k_m <= 4.8e-6) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t_m, 1.5) * (Math.pow(t_m, 1.5) / l)) / l));
	} else if (k_m <= 1.9e+94) {
		tmp = t_2 * (l / (2.0 + ((k_m / t_m) / (t_m / k_m))));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow((t_m * Math.cbrt((k_m / Math.pow(l, 2.0)))), 3.0));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k_m)) * Float64(l / tan(k_m)))
	tmp = 0.0
	if (k_m <= 3.3e-167)
		tmp = Float64(t_2 * Float64(l * 0.5));
	elseif (k_m <= 4.8e-6)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t_m ^ 1.5) * Float64((t_m ^ 1.5) / l)) / l)));
	elseif (k_m <= 1.9e+94)
		tmp = Float64(t_2 * Float64(l / Float64(2.0 + Float64(Float64(k_m / t_m) / Float64(t_m / k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(t_m * cbrt(Float64(k_m / (l ^ 2.0)))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 3.3e-167], N[(t$95$2 * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.8e-6], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.9e+94], N[(t$95$2 * N[(l / N[(2.0 + N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 3.29999999999999995e-167

   1. Initial program 57.5%

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

   3. Simplified 52.1%

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

      1. associate-*r* 58.8%

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

      2. *-un-lft-identity 58.8%

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

      3. times-frac 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. /-rgt-identity 59.2%

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

      2. associate-*l/ 59.2%

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

      3. associate-*r* 62.9%

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

      4. times-frac 65.1%

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

      5. associate-/r* 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in k around 0 63.5%

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

       1. *-commutative 63.5%

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

   11. Simplified 63.5%

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

   if 3.29999999999999995e-167 < k < 4.7999999999999998e-6

   1. Initial program 66.4%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in k around 0 78.5%

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

      1. sqr-pow 39.8%

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

      2. *-un-lft-identity 39.8%

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

      3. times-frac 44.3%

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

      4. metadata-eval 44.3%

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

      5. metadata-eval 44.3%

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

   7. Applied egg-rr 44.3%

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

   if 4.7999999999999998e-6 < k < 1.8999999999999998e94

   1. Initial program 75.6%

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

   3. Simplified 75.7%

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

      1. associate-*r* 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. times-frac 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. /-rgt-identity 87.4%

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

      2. associate-*l/ 87.4%

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

      3. associate-*r* 87.4%

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

      4. times-frac 87.4%

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

      5. associate-/r* 87.3%

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

   8. Simplified 87.3%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

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

   10. Applied egg-rr 87.3%

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

   if 1.8999999999999998e94 < k

   1. Initial program 42.3%

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

   3. Simplified 42.4%

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

      1. add-cube-cbrt 42.3%

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

      2. pow3 42.3%

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

      3. *-commutative 42.3%

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

      4. cbrt-prod 42.3%

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

      5. cbrt-div 42.3%

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

      6. rem-cbrt-cube 53.0%

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

      7. cbrt-prod 58.0%

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

      8. pow2 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. *-commutative 58.0%

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

   8. Simplified 58.0%

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

      1. unpow2 58.0%

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

      2. clear-num 58.0%

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

      3. un-div-inv 58.0%

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

   10. Applied egg-rr 58.0%

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

   11. Taylor expanded in k around 0 58.6%

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

   12. Taylor expanded in k around 0 56.2%

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

       1. *-commutative 56.2%

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

   14. Simplified 56.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{1.5} \cdot \frac{{t}^{1.5}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\ell}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{-288}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k\_m \leq 1120000000000:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k_m 2.0))))
   (*
    t_s
    (if (<= k_m 6.6e-288)
      (/ 2.0 (* t_2 (/ (/ (pow t_m 3.0) l) l)))
      (if (<= k_m 3.3e-167)
        (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* l l)))))
        (if (<= k_m 1120000000000.0)
          (/ 2.0 (* t_2 (/ (pow (/ t_m (cbrt l)) 3.0) l)))
          (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 2.0 * pow(k_m, 2.0);
	double tmp;
	if (k_m <= 6.6e-288) {
		tmp = 2.0 / (t_2 * ((pow(t_m, 3.0) / l) / l));
	} else if (k_m <= 3.3e-167) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * l))));
	} else if (k_m <= 1120000000000.0) {
		tmp = 2.0 / (t_2 * (pow((t_m / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 2.0 * Math.pow(k_m, 2.0);
	double tmp;
	if (k_m <= 6.6e-288) {
		tmp = 2.0 / (t_2 * ((Math.pow(t_m, 3.0) / l) / l));
	} else if (k_m <= 3.3e-167) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l))));
	} else if (k_m <= 1120000000000.0) {
		tmp = 2.0 / (t_2 * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(2.0 * (k_m ^ 2.0))
	tmp = 0.0
	if (k_m <= 6.6e-288)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t_m ^ 3.0) / l) / l)));
	elseif (k_m <= 3.3e-167)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	elseif (k_m <= 1120000000000.0)
		tmp = Float64(2.0 / Float64(t_2 * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6.6e-288], N[(2.0 / N[(t$95$2 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.3e-167], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1120000000000.0], N[(2.0 / N[(t$95$2 * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 6.59999999999999976e-288

   1. Initial program 54.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in k around 0 53.7%

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

   if 6.59999999999999976e-288 < k < 3.29999999999999995e-167

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in k around 0 72.8%

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

   if 3.29999999999999995e-167 < k < 1.12e12

   1. Initial program 63.6%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in k around 0 75.2%

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

      1. add-cube-cbrt 75.1%

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

      2. pow3 75.1%

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

      3. cbrt-div 75.0%

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

      4. rem-cbrt-cube 79.3%

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

   7. Applied egg-rr 79.3%

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

   if 1.12e12 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 66.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-288}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 1120000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.22 \cdot 10^{-288}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 2.9 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot {\left(t\_m \cdot \sqrt[3]{\frac{k\_m}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 6000000000000:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k_m 2.0))))
   (*
    t_s
    (if (<= k_m 1.22e-288)
      (/ 2.0 (* t_2 (/ (/ (pow t_m 3.0) l) l)))
      (if (<= k_m 2.9e-146)
        (/ 2.0 (* (* k_m 2.0) (pow (* t_m (cbrt (/ k_m (pow l 2.0)))) 3.0)))
        (if (<= k_m 6000000000000.0)
          (/ 2.0 (* t_2 (/ (pow (/ t_m (cbrt l)) 3.0) l)))
          (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 2.0 * pow(k_m, 2.0);
	double tmp;
	if (k_m <= 1.22e-288) {
		tmp = 2.0 / (t_2 * ((pow(t_m, 3.0) / l) / l));
	} else if (k_m <= 2.9e-146) {
		tmp = 2.0 / ((k_m * 2.0) * pow((t_m * cbrt((k_m / pow(l, 2.0)))), 3.0));
	} else if (k_m <= 6000000000000.0) {
		tmp = 2.0 / (t_2 * (pow((t_m / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 2.0 * Math.pow(k_m, 2.0);
	double tmp;
	if (k_m <= 1.22e-288) {
		tmp = 2.0 / (t_2 * ((Math.pow(t_m, 3.0) / l) / l));
	} else if (k_m <= 2.9e-146) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow((t_m * Math.cbrt((k_m / Math.pow(l, 2.0)))), 3.0));
	} else if (k_m <= 6000000000000.0) {
		tmp = 2.0 / (t_2 * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(2.0 * (k_m ^ 2.0))
	tmp = 0.0
	if (k_m <= 1.22e-288)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t_m ^ 3.0) / l) / l)));
	elseif (k_m <= 2.9e-146)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(t_m * cbrt(Float64(k_m / (l ^ 2.0)))) ^ 3.0)));
	elseif (k_m <= 6000000000000.0)
		tmp = Float64(2.0 / Float64(t_2 * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.22e-288], N[(2.0 / N[(t$95$2 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.9e-146], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[(k$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6000000000000.0], N[(2.0 / N[(t$95$2 * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 1.22e-288

   1. Initial program 54.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in k around 0 53.7%

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

   if 1.22e-288 < k < 2.90000000000000011e-146

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

      1. add-cube-cbrt 71.5%

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

      2. pow3 71.5%

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

      3. *-commutative 71.5%

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

      4. cbrt-prod 71.4%

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

      5. cbrt-div 71.4%

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

      6. rem-cbrt-cube 81.2%

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

      7. cbrt-prod 90.3%

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

      8. pow2 90.3%

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

   6. Applied egg-rr 90.3%

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

      1. *-commutative 90.3%

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

   8. Simplified 90.3%

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

      1. unpow2 90.3%

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

      2. clear-num 90.3%

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

      3. un-div-inv 90.3%

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

   10. Applied egg-rr 90.3%

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

   11. Taylor expanded in k around 0 75.2%

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

   12. Taylor expanded in k around 0 75.2%

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

       1. *-commutative 75.2%

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

   14. Simplified 75.2%

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

   if 2.90000000000000011e-146 < k < 6e12

   1. Initial program 63.0%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in k around 0 76.4%

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

      1. add-cube-cbrt 76.3%

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

      2. pow3 76.3%

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

      3. cbrt-div 76.2%

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

      4. rem-cbrt-cube 78.7%

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

   7. Applied egg-rr 78.7%

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

   if 6e12 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 66.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{-288}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(t \cdot \sqrt[3]{\frac{k}{{\ell}^{2}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 6000000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-288}:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k\_m \leq 1.25 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k\_m \leq 1120000000000:\\ \;\;\;\;\frac{2}{t\_2 \cdot \frac{{t\_m}^{2} \cdot \frac{t\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow k_m 2.0))))
   (*
    t_s
    (if (<= k_m 2.4e-288)
      (/ 2.0 (* t_2 (/ (/ (pow t_m 3.0) l) l)))
      (if (<= k_m 1.25e-167)
        (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* l l)))))
        (if (<= k_m 1120000000000.0)
          (/ 2.0 (* t_2 (/ (* (pow t_m 2.0) (/ t_m l)) l)))
          (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 2.0 * pow(k_m, 2.0);
	double tmp;
	if (k_m <= 2.4e-288) {
		tmp = 2.0 / (t_2 * ((pow(t_m, 3.0) / l) / l));
	} else if (k_m <= 1.25e-167) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * l))));
	} else if (k_m <= 1120000000000.0) {
		tmp = 2.0 / (t_2 * ((pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 * (k_m ** 2.0d0)
    if (k_m <= 2.4d-288) then
        tmp = 2.0d0 / (t_2 * (((t_m ** 3.0d0) / l) / l))
    else if (k_m <= 1.25d-167) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t_m ** 3.0d0) / (l * l))))
    else if (k_m <= 1120000000000.0d0) then
        tmp = 2.0d0 / (t_2 * (((t_m ** 2.0d0) * (t_m / l)) / l))
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 2.0 * Math.pow(k_m, 2.0);
	double tmp;
	if (k_m <= 2.4e-288) {
		tmp = 2.0 / (t_2 * ((Math.pow(t_m, 3.0) / l) / l));
	} else if (k_m <= 1.25e-167) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l))));
	} else if (k_m <= 1120000000000.0) {
		tmp = 2.0 / (t_2 * ((Math.pow(t_m, 2.0) * (t_m / l)) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = 2.0 * math.pow(k_m, 2.0)
	tmp = 0
	if k_m <= 2.4e-288:
		tmp = 2.0 / (t_2 * ((math.pow(t_m, 3.0) / l) / l))
	elif k_m <= 1.25e-167:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l))))
	elif k_m <= 1120000000000.0:
		tmp = 2.0 / (t_2 * ((math.pow(t_m, 2.0) * (t_m / l)) / l))
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / (math.pow(l, 2.0) * math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(2.0 * (k_m ^ 2.0))
	tmp = 0.0
	if (k_m <= 2.4e-288)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t_m ^ 3.0) / l) / l)));
	elseif (k_m <= 1.25e-167)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	elseif (k_m <= 1120000000000.0)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = 2.0 * (k_m ^ 2.0);
	tmp = 0.0;
	if (k_m <= 2.4e-288)
		tmp = 2.0 / (t_2 * (((t_m ^ 3.0) / l) / l));
	elseif (k_m <= 1.25e-167)
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t_m ^ 3.0) / (l * l))));
	elseif (k_m <= 1120000000000.0)
		tmp = 2.0 / (t_2 * (((t_m ^ 2.0) * (t_m / l)) / l));
	else
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.4e-288], N[(2.0 / N[(t$95$2 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.25e-167], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1120000000000.0], N[(2.0 / N[(t$95$2 * 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[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 2.3999999999999998e-288

   1. Initial program 54.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in k around 0 53.7%

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

   if 2.3999999999999998e-288 < k < 1.25000000000000005e-167

   1. Initial program 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in k around 0 72.8%

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

   if 1.25000000000000005e-167 < k < 1.12e12

   1. Initial program 63.6%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in k around 0 75.2%

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

      1. unpow3 75.2%

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

      2. *-un-lft-identity 75.2%

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

      3. times-frac 77.4%

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

      4. pow2 77.4%

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

   7. Applied egg-rr 77.4%

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

   if 1.12e12 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 66.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-288}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 1120000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;k\_m \leq 1120000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t\_m}^{1.5} \cdot \frac{{t\_m}^{1.5}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.3e-167)
    (* (* (/ (/ 2.0 (pow t_m 3.0)) (sin k_m)) (/ l (tan k_m))) (* l 0.5))
    (if (<= k_m 1120000000000.0)
      (/
       2.0
       (* (* 2.0 (pow k_m 2.0)) (/ (* (pow t_m 1.5) (/ (pow t_m 1.5) l)) l)))
      (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.3e-167) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k_m)) * (l / tan(k_m))) * (l * 0.5);
	} else if (k_m <= 1120000000000.0) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t_m, 1.5) * (pow(t_m, 1.5) / l)) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.3d-167) then
        tmp = (((2.0d0 / (t_m ** 3.0d0)) / sin(k_m)) * (l / tan(k_m))) * (l * 0.5d0)
    else if (k_m <= 1120000000000.0d0) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t_m ** 1.5d0) * ((t_m ** 1.5d0) / l)) / l))
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 3.3e-167) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k_m)) * (l / Math.tan(k_m))) * (l * 0.5);
	} else if (k_m <= 1120000000000.0) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t_m, 1.5) * (Math.pow(t_m, 1.5) / l)) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.3e-167:
		tmp = (((2.0 / math.pow(t_m, 3.0)) / math.sin(k_m)) * (l / math.tan(k_m))) * (l * 0.5)
	elif k_m <= 1120000000000.0:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t_m, 1.5) * (math.pow(t_m, 1.5) / l)) / l))
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / (math.pow(l, 2.0) * math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.3e-167)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k_m)) * Float64(l / tan(k_m))) * Float64(l * 0.5));
	elseif (k_m <= 1120000000000.0)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t_m ^ 1.5) * Float64((t_m ^ 1.5) / l)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.3e-167)
		tmp = (((2.0 / (t_m ^ 3.0)) / sin(k_m)) * (l / tan(k_m))) * (l * 0.5);
	elseif (k_m <= 1120000000000.0)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t_m ^ 1.5) * ((t_m ^ 1.5) / l)) / l));
	else
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.3e-167], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1120000000000.0], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.29999999999999995e-167

   1. Initial program 57.5%

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

   3. Simplified 52.1%

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

      1. associate-*r* 58.8%

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

      2. *-un-lft-identity 58.8%

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

      3. times-frac 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. /-rgt-identity 59.2%

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

      2. associate-*l/ 59.2%

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

      3. associate-*r* 62.9%

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

      4. times-frac 65.1%

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

      5. associate-/r* 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in k around 0 63.5%

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

       1. *-commutative 63.5%

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

   11. Simplified 63.5%

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

   if 3.29999999999999995e-167 < k < 1.12e12

   1. Initial program 63.6%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in k around 0 75.2%

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

      1. sqr-pow 38.0%

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

      2. *-un-lft-identity 38.0%

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

      3. times-frac 42.4%

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

      4. metadata-eval 42.4%

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

      5. metadata-eval 42.4%

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

   7. Applied egg-rr 42.4%

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

   if 1.12e12 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 66.1%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;k \leq 1120000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{1.5} \cdot \frac{{t}^{1.5}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\left(\frac{\frac{2}{{t\_m}^{3}}}{\sin k\_m} \cdot \frac{\ell}{\tan k\_m}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;k\_m \leq 1040000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.3e-167)
    (* (* (/ (/ 2.0 (pow t_m 3.0)) (sin k_m)) (/ l (tan k_m))) (* l 0.5))
    (if (<= k_m 1040000000000.0)
      (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l)))
      (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.3e-167) {
		tmp = (((2.0 / pow(t_m, 3.0)) / sin(k_m)) * (l / tan(k_m))) * (l * 0.5);
	} else if (k_m <= 1040000000000.0) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 3.3e-167) {
		tmp = (((2.0 / Math.pow(t_m, 3.0)) / Math.sin(k_m)) * (l / Math.tan(k_m))) * (l * 0.5);
	} else if (k_m <= 1040000000000.0) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.3e-167)
		tmp = Float64(Float64(Float64(Float64(2.0 / (t_m ^ 3.0)) / sin(k_m)) * Float64(l / tan(k_m))) * Float64(l * 0.5));
	elseif (k_m <= 1040000000000.0)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.3e-167], N[(N[(N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1040000000000.0], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 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[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.29999999999999995e-167

   1. Initial program 57.5%

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

   3. Simplified 52.1%

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

      1. associate-*r* 58.8%

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

      2. *-un-lft-identity 58.8%

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

      3. times-frac 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. /-rgt-identity 59.2%

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

      2. associate-*l/ 59.2%

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

      3. associate-*r* 62.9%

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

      4. times-frac 65.1%

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

      5. associate-/r* 65.2%

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

   8. Simplified 65.2%

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

   9. Taylor expanded in k around 0 63.5%

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

       1. *-commutative 63.5%

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

   11. Simplified 63.5%

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

   if 3.29999999999999995e-167 < k < 1.04e12

   1. Initial program 63.6%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in k around 0 75.2%

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

      1. add-cube-cbrt 75.1%

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

      2. pow3 75.1%

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

      3. cbrt-div 75.0%

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

      4. rem-cbrt-cube 79.3%

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

   7. Applied egg-rr 79.3%

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

   if 1.04e12 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 66.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \left(\ell \cdot 0.5\right)\\ \mathbf{elif}\;k \leq 1040000000000:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left({k\_m}^{4} \cdot {\ell}^{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-91)
    (/ 2.0 (* t_m (* (pow k_m 4.0) (pow l -2.0))))
    (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* l l))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3e-91) {
		tmp = 2.0 / (t_m * (pow(k_m, 4.0) * pow(l, -2.0)));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3d-91) then
        tmp = 2.0d0 / (t_m * ((k_m ** 4.0d0) * (l ** (-2.0d0))))
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t_m ** 3.0d0) / (l * l))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 3e-91) {
		tmp = 2.0 / (t_m * (Math.pow(k_m, 4.0) * Math.pow(l, -2.0)));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 3e-91:
		tmp = 2.0 / (t_m * (math.pow(k_m, 4.0) * math.pow(l, -2.0)))
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 3e-91)
		tmp = Float64(2.0 / Float64(t_m * Float64((k_m ^ 4.0) * (l ^ -2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 3e-91)
		tmp = 2.0 / (t_m * ((k_m ^ 4.0) * (l ^ -2.0)));
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t_m ^ 3.0) / (l * l))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-91], N[(2.0 / N[(t$95$m * N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.0000000000000002e-91

   1. Initial program 51.5%

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

   3. Taylor expanded in t around 0 62.1%

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

      1. associate-*r* 62.1%

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

   5. Simplified 62.1%

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

   6. Taylor expanded in k around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-/l* 51.6%

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

   8. Simplified 51.6%

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

      1. *-un-lft-identity 51.6%

         \[\leadsto \frac{2}{t \cdot \color{blue}{\left(1 \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]

      2. div-inv 51.6%

         \[\leadsto \frac{2}{t \cdot \left(1 \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)} \]

      3. pow-flip 51.6%

         \[\leadsto \frac{2}{t \cdot \left(1 \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)} \]

      4. metadata-eval 51.6%

         \[\leadsto \frac{2}{t \cdot \left(1 \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)} \]

   10. Applied egg-rr 51.6%

       \[\leadsto \frac{2}{t \cdot \color{blue}{\left(1 \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-lft-identity 51.6%

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

   12. Simplified 51.6%

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

   if 3.0000000000000002e-91 < t

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in k around 0 62.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.15e+14)
    (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (/ (pow t_m 3.0) l) l)))
    (/ 2.0 (/ (* t_m (pow k_m 4.0)) (pow l 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e+14) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.15d+14) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t_m ** 3.0d0) / l) / l))
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2.15e+14) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.15e+14:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t_m, 3.0) / l) / l))
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.15e+14)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.15e+14)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t_m ^ 3.0) / l) / l));
	else
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.15e+14], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.15e14

   1. Initial program 58.9%

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

   3. Simplified 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 59.4%

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

   if 2.15e14 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 62.2%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{2 \cdot {k\_m}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.9e+15)
    (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* 2.0 (pow k_m 2.0)) l)))
    (/ 2.0 (/ (* t_m (pow k_m 4.0)) (pow l 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.9e+15) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((2.0 * pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.9d+15) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * ((2.0d0 * (k_m ** 2.0d0)) / l))
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 3.9e+15) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((2.0 * Math.pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.9e+15:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * ((2.0 * math.pow(k_m, 2.0)) / l))
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.9e+15)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(2.0 * (k_m ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.9e+15)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * ((2.0 * (k_m ^ 2.0)) / l));
	else
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.9e+15], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.9e15

   1. Initial program 58.9%

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

   3. Simplified 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 59.4%

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

      1. associate-*l/ 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. associate-/l* 60.2%

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

   9. Simplified 60.2%

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

   if 3.9e15 < k

   1. Initial program 54.1%

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

   3. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in k around 0 62.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{t\_m \cdot \left({k\_m}^{4} \cdot {\ell}^{-2}\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* t_m (* (pow k_m 4.0) (pow l -2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / (t_m * (pow(k_m, 4.0) * pow(l, -2.0))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / (t_m * ((k_m ** 4.0d0) * (l ** (-2.0d0)))))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / (t_m * (Math.pow(k_m, 4.0) * Math.pow(l, -2.0))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / (t_m * (math.pow(k_m, 4.0) * math.pow(l, -2.0))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(t_m * Float64((k_m ^ 4.0) * (l ^ -2.0)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / (t_m * ((k_m ^ 4.0) * (l ^ -2.0))));
end

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in t around 0 60.5%

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

   1. associate-*r* 60.5%

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

5. Simplified 60.5%

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

6. Taylor expanded in k around 0 51.0%

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

   1. *-commutative 51.0%

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

   2. associate-/l* 50.7%

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

8. Simplified 50.7%

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

   1. *-un-lft-identity 50.7%

      \[\leadsto \frac{2}{t \cdot \color{blue}{\left(1 \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]

   2. div-inv 50.7%

      \[\leadsto \frac{2}{t \cdot \left(1 \cdot \color{blue}{\left({k}^{4} \cdot \frac{1}{{\ell}^{2}}\right)}\right)} \]

   3. pow-flip 50.7%

      \[\leadsto \frac{2}{t \cdot \left(1 \cdot \left({k}^{4} \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)} \]

   4. metadata-eval 50.7%

      \[\leadsto \frac{2}{t \cdot \left(1 \cdot \left({k}^{4} \cdot {\ell}^{\color{blue}{-2}}\right)\right)} \]

10. Applied egg-rr 50.7%

    \[\leadsto \frac{2}{t \cdot \color{blue}{\left(1 \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)\right)}} \]
11. Step-by-step derivation

    1. *-lft-identity 50.7%

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

12. Simplified 50.7%

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

13. Final simplification 50.7%

    \[\leadsto \frac{2}{t \cdot \left({k}^{4} \cdot {\ell}^{-2}\right)} \]
14. Add Preprocessing

Alternative 23: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{t\_m \cdot \frac{{k\_m}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* t_m (/ (pow k_m 4.0) (pow l 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / (t_m * (pow(k_m, 4.0) / pow(l, 2.0))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / (t_m * ((k_m ** 4.0d0) / (l ** 2.0d0))))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / (t_m * (Math.pow(k_m, 4.0) / Math.pow(l, 2.0))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / (t_m * (math.pow(k_m, 4.0) / math.pow(l, 2.0))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(t_m * Float64((k_m ^ 4.0) / (l ^ 2.0)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / (t_m * ((k_m ^ 4.0) / (l ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in t around 0 60.5%

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

   1. associate-*r* 60.5%

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

5. Simplified 60.5%

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

6. Taylor expanded in k around 0 51.0%

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

   1. *-commutative 51.0%

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

   2. associate-/l* 50.7%

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

8. Simplified 50.7%

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

9. Final simplification 50.7%

   \[\leadsto \frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]
10. Add Preprocessing

Alternative 24: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{k\_m}^{4} \cdot \frac{t\_m}{{\ell}^{2}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* (pow k_m 4.0) (/ t_m (pow l 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / (pow(k_m, 4.0) * (t_m / pow(l, 2.0))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / ((k_m ** 4.0d0) * (t_m / (l ** 2.0d0))))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / (Math.pow(k_m, 4.0) * (t_m / Math.pow(l, 2.0))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / (math.pow(k_m, 4.0) * (t_m / math.pow(l, 2.0))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t_m / (l ^ 2.0)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((k_m ^ 4.0) * (t_m / (l ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in t around 0 60.5%

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

   1. associate-*r* 60.5%

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

5. Simplified 60.5%

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

6. Taylor expanded in k around 0 51.0%

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

   1. associate-/l* 51.3%

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

8. Simplified 51.3%

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

9. Final simplification 51.3%

   \[\leadsto \frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))