Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 78.3%

Time: 33.0s

Alternatives: 18

Speedup: 2.0×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 35.2% 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: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k_m \cdot \tan k_m\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(\left(\frac{k_m}{\ell} \cdot \sqrt{t_m}\right) \cdot \sqrt{t_2}\right)}^{4}}}\\ \mathbf{elif}\;k_m \leq 2.3 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{{\ell}^{2} \cdot \cos k_m}{t_m \cdot {\sin k_m}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\frac{k_m}{t_m}}\right)}^{2} \cdot \frac{t_m}{\sqrt[3]{\frac{{\ell}^{2}}{t_2}}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (sin k_m) (tan k_m))))
   (*
    t_s
    (if (<= k_m 1.35e-145)
      (sqrt (/ 4.0 (pow (* (* (/ k_m l) (sqrt t_m)) (sqrt t_2)) 4.0)))
      (if (<= k_m 2.3e+124)
        (/
         2.0
         (/
          (pow k_m 2.0)
          (/ (* (pow l 2.0) (cos k_m)) (* t_m (pow (sin k_m) 2.0)))))
        (/
         2.0
         (pow
          (* (pow (cbrt (/ k_m t_m)) 2.0) (/ t_m (cbrt (/ (pow l 2.0) t_2))))
          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 = sin(k_m) * tan(k_m);
	double tmp;
	if (k_m <= 1.35e-145) {
		tmp = sqrt((4.0 / pow((((k_m / l) * sqrt(t_m)) * sqrt(t_2)), 4.0)));
	} else if (k_m <= 2.3e+124) {
		tmp = 2.0 / (pow(k_m, 2.0) / ((pow(l, 2.0) * cos(k_m)) / (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / pow((pow(cbrt((k_m / t_m)), 2.0) * (t_m / cbrt((pow(l, 2.0) / t_2)))), 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.sin(k_m) * Math.tan(k_m);
	double tmp;
	if (k_m <= 1.35e-145) {
		tmp = Math.sqrt((4.0 / Math.pow((((k_m / l) * Math.sqrt(t_m)) * Math.sqrt(t_2)), 4.0)));
	} else if (k_m <= 2.3e+124) {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k_m)) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / Math.pow((Math.pow(Math.cbrt((k_m / t_m)), 2.0) * (t_m / Math.cbrt((Math.pow(l, 2.0) / t_2)))), 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(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (k_m <= 1.35e-145)
		tmp = sqrt(Float64(4.0 / (Float64(Float64(Float64(k_m / l) * sqrt(t_m)) * sqrt(t_2)) ^ 4.0)));
	elseif (k_m <= 2.3e+124)
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64((cbrt(Float64(k_m / t_m)) ^ 2.0) * Float64(t_m / cbrt(Float64((l ^ 2.0) / t_2)))) ^ 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[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.35e-145], N[Sqrt[N[(4.0 / N[Power[N[(N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k$95$m, 2.3e+124], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Power[N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[Power[N[(N[Power[l, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.35e-145

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. associate-/r* 36.2%

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

      3. sub-neg 36.2%

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

      4. distribute-rgt-in 29.7%

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

      5. unpow2 29.7%

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

      6. times-frac 22.6%

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

      7. sqr-neg 22.6%

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

      8. times-frac 29.7%

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

      9. unpow2 29.7%

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

      10. distribute-rgt-in 36.2%

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

      11. +-commutative 36.2%

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

      12. associate-+l+ 40.8%

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

   3. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 17.5%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}}} \]

      2. pow-sqr 17.5%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{\left(2 \cdot 2\right)}}}} \]

      3. associate-*r* 17.5%

         \[\leadsto \sqrt{\frac{4}{{\color{blue}{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}}^{\left(2 \cdot 2\right)}}} \]

      4. associate-*r/ 18.8%

         \[\leadsto \sqrt{\frac{4}{{\left(\color{blue}{\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\left(2 \cdot 2\right)}}} \]

      5. metadata-eval 18.8%

         \[\leadsto \sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\color{blue}{4}}}} \]

   8. Simplified 18.8%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{4}}}} \]

   9. Taylor expanded in k around 0 25.9%

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

   if 1.35e-145 < k < 2.29999999999999985e124

   1. Initial program 17.8%

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

   3. Taylor expanded in t around 0 87.3%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

   if 2.29999999999999985e124 < 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 68.7%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(\left(\frac{k}{\ell} \cdot \sqrt{t}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{4}}}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\frac{k}{t}}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\frac{{\ell}^{2}}{\sin k \cdot \tan k}}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 10^{-145}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(\left(\frac{k_m}{\ell} \cdot \sqrt{t_m}\right) \cdot \sqrt{\sin k_m \cdot \tan k_m}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{{\ell}^{2} \cdot \cos k_m}{t_m \cdot {\sin k_m}^{2}}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1e-145)
    (sqrt
     (/
      4.0
      (pow (* (* (/ k_m l) (sqrt t_m)) (sqrt (* (sin k_m) (tan k_m)))) 4.0)))
    (/
     2.0
     (/
      (pow k_m 2.0)
      (/ (* (pow l 2.0) (cos k_m)) (* t_m (pow (sin k_m) 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 <= 1e-145) {
		tmp = sqrt((4.0 / pow((((k_m / l) * sqrt(t_m)) * sqrt((sin(k_m) * tan(k_m)))), 4.0)));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) / ((pow(l, 2.0) * cos(k_m)) / (t_m * pow(sin(k_m), 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 <= 1d-145) then
        tmp = sqrt((4.0d0 / ((((k_m / l) * sqrt(t_m)) * sqrt((sin(k_m) * tan(k_m)))) ** 4.0d0)))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) / (((l ** 2.0d0) * cos(k_m)) / (t_m * (sin(k_m) ** 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 <= 1e-145) {
		tmp = Math.sqrt((4.0 / Math.pow((((k_m / l) * Math.sqrt(t_m)) * Math.sqrt((Math.sin(k_m) * Math.tan(k_m)))), 4.0)));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k_m)) / (t_m * Math.pow(Math.sin(k_m), 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 <= 1e-145:
		tmp = math.sqrt((4.0 / math.pow((((k_m / l) * math.sqrt(t_m)) * math.sqrt((math.sin(k_m) * math.tan(k_m)))), 4.0)))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) / ((math.pow(l, 2.0) * math.cos(k_m)) / (t_m * math.pow(math.sin(k_m), 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 <= 1e-145)
		tmp = sqrt(Float64(4.0 / (Float64(Float64(Float64(k_m / l) * sqrt(t_m)) * sqrt(Float64(sin(k_m) * tan(k_m)))) ^ 4.0)));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(t_m * (sin(k_m) ^ 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 <= 1e-145)
		tmp = sqrt((4.0 / ((((k_m / l) * sqrt(t_m)) * sqrt((sin(k_m) * tan(k_m)))) ^ 4.0)));
	else
		tmp = 2.0 / ((k_m ^ 2.0) / (((l ^ 2.0) * cos(k_m)) / (t_m * (sin(k_m) ^ 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, 1e-145], N[Sqrt[N[(4.0 / N[Power[N[(N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999915e-146

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. associate-/r* 36.2%

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

      3. sub-neg 36.2%

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

      4. distribute-rgt-in 29.7%

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

      5. unpow2 29.7%

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

      6. times-frac 22.6%

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

      7. sqr-neg 22.6%

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

      8. times-frac 29.7%

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

      9. unpow2 29.7%

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

      10. distribute-rgt-in 36.2%

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

      11. +-commutative 36.2%

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

      12. associate-+l+ 40.8%

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

   3. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 17.5%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 17.5%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}}} \]

      2. pow-sqr 17.5%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{\left(2 \cdot 2\right)}}}} \]

      3. associate-*r* 17.5%

         \[\leadsto \sqrt{\frac{4}{{\color{blue}{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}}^{\left(2 \cdot 2\right)}}} \]

      4. associate-*r/ 18.8%

         \[\leadsto \sqrt{\frac{4}{{\left(\color{blue}{\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\left(2 \cdot 2\right)}}} \]

      5. metadata-eval 18.8%

         \[\leadsto \sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\color{blue}{4}}}} \]

   8. Simplified 18.8%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{4}}}} \]

   9. Taylor expanded in k around 0 25.9%

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

   if 9.99999999999999915e-146 < k

   1. Initial program 24.2%

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

   3. Taylor expanded in t around 0 77.5%

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

      1. associate-/l* 77.7%

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

   5. Simplified 77.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-145}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(\left(\frac{k}{\ell} \cdot \sqrt{t}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\log \left({\left({\left(e^{{k_m}^{4}}\right)}^{t_m}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{{\ell}^{2} \cdot \cos k_m}{t_m \cdot {\sin k_m}^{2}}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.7e-146)
    (/ 2.0 (log (pow (pow (exp (pow k_m 4.0)) t_m) (pow l -2.0))))
    (/
     2.0
     (/
      (pow k_m 2.0)
      (/ (* (pow l 2.0) (cos k_m)) (* t_m (pow (sin k_m) 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 <= 2.7e-146) {
		tmp = 2.0 / log(pow(pow(exp(pow(k_m, 4.0)), t_m), pow(l, -2.0)));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) / ((pow(l, 2.0) * cos(k_m)) / (t_m * pow(sin(k_m), 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 <= 2.7d-146) then
        tmp = 2.0d0 / log(((exp((k_m ** 4.0d0)) ** t_m) ** (l ** (-2.0d0))))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) / (((l ** 2.0d0) * cos(k_m)) / (t_m * (sin(k_m) ** 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 <= 2.7e-146) {
		tmp = 2.0 / Math.log(Math.pow(Math.pow(Math.exp(Math.pow(k_m, 4.0)), t_m), Math.pow(l, -2.0)));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k_m)) / (t_m * Math.pow(Math.sin(k_m), 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 <= 2.7e-146:
		tmp = 2.0 / math.log(math.pow(math.pow(math.exp(math.pow(k_m, 4.0)), t_m), math.pow(l, -2.0)))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) / ((math.pow(l, 2.0) * math.cos(k_m)) / (t_m * math.pow(math.sin(k_m), 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 <= 2.7e-146)
		tmp = Float64(2.0 / log(((exp((k_m ^ 4.0)) ^ t_m) ^ (l ^ -2.0))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(t_m * (sin(k_m) ^ 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 <= 2.7e-146)
		tmp = 2.0 / log(((exp((k_m ^ 4.0)) ^ t_m) ^ (l ^ -2.0)));
	else
		tmp = 2.0 / ((k_m ^ 2.0) / (((l ^ 2.0) * cos(k_m)) / (t_m * (sin(k_m) ^ 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, 2.7e-146], N[(2.0 / N[Log[N[Power[N[Power[N[Exp[N[Power[k$95$m, 4.0], $MachinePrecision]], $MachinePrecision], t$95$m], $MachinePrecision], N[Power[l, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999995e-146

   1. Initial program 36.2%

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

   3. Taylor expanded in k around 0 67.6%

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

      1. add-log-exp 40.3%

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

      2. div-inv 40.3%

         \[\leadsto \frac{2}{\log \left(e^{\color{blue}{\left({k}^{4} \cdot t\right) \cdot \frac{1}{{\ell}^{2}}}}\right)} \]

      3. *-commutative 40.3%

         \[\leadsto \frac{2}{\log \left(e^{\color{blue}{\left(t \cdot {k}^{4}\right)} \cdot \frac{1}{{\ell}^{2}}}\right)} \]

      4. exp-prod 41.8%

         \[\leadsto \frac{2}{\log \color{blue}{\left({\left(e^{t \cdot {k}^{4}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)}} \]

      5. expm1-log1p-u 30.9%

         \[\leadsto \frac{2}{\log \left({\left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot {k}^{4}\right)\right)}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)} \]

      6. expm1-log1p-u 41.8%

         \[\leadsto \frac{2}{\log \left({\left(e^{\color{blue}{t \cdot {k}^{4}}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)} \]

      7. *-commutative 41.8%

         \[\leadsto \frac{2}{\log \left({\left(e^{\color{blue}{{k}^{4} \cdot t}}\right)}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)} \]

      8. exp-prod 42.4%

         \[\leadsto \frac{2}{\log \left({\color{blue}{\left({\left(e^{{k}^{4}}\right)}^{t}\right)}}^{\left(\frac{1}{{\ell}^{2}}\right)}\right)} \]

      9. pow-flip 42.4%

         \[\leadsto \frac{2}{\log \left({\left({\left(e^{{k}^{4}}\right)}^{t}\right)}^{\color{blue}{\left({\ell}^{\left(-2\right)}\right)}}\right)} \]

      10. metadata-eval 42.4%

          \[\leadsto \frac{2}{\log \left({\left({\left(e^{{k}^{4}}\right)}^{t}\right)}^{\left({\ell}^{\color{blue}{-2}}\right)}\right)} \]

   5. Applied egg-rr 42.4%

      \[\leadsto \frac{2}{\color{blue}{\log \left({\left({\left(e^{{k}^{4}}\right)}^{t}\right)}^{\left({\ell}^{-2}\right)}\right)}} \]

   if 2.69999999999999995e-146 < k

   1. Initial program 24.2%

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

   3. Taylor expanded in t around 0 77.5%

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

      1. associate-/l* 77.7%

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

   5. Simplified 77.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{\log \left({\left({\left(e^{{k}^{4}}\right)}^{t}\right)}^{\left({\ell}^{-2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{t_m \cdot {k_m}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k_m}{t_2}\\ \mathbf{elif}\;t_m \leq 4.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t_m}^{2}}{\frac{\ell}{\sin k_m}} \cdot \frac{t_m}{\frac{\ell}{\tan k_m}}}}{{\left(\frac{k_m}{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{t_2}}{{k_m}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (sin k_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.4e-42)
      (* (/ 2.0 (* t_m (pow k_m 2.0))) (/ (* (pow l 2.0) (cos k_m)) t_2))
      (if (<= t_m 4.5e+133)
        (/
         (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k_m))) (/ t_m (/ l (tan k_m)))))
         (pow (/ k_m t_m) 2.0))
        (*
         2.0
         (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) t_2)) (pow k_m 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 t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (t_m <= 1.4e-42) {
		tmp = (2.0 / (t_m * pow(k_m, 2.0))) * ((pow(l, 2.0) * cos(k_m)) / t_2);
	} else if (t_m <= 4.5e+133) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / pow((k_m / t_m), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((cos(k_m) / t_m) / t_2)) / pow(k_m, 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) :: t_2
    real(8) :: tmp
    t_2 = sin(k_m) ** 2.0d0
    if (t_m <= 1.4d-42) then
        tmp = (2.0d0 / (t_m * (k_m ** 2.0d0))) * (((l ** 2.0d0) * cos(k_m)) / t_2)
    else if (t_m <= 4.5d+133) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / ((k_m / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / t_2)) / (k_m ** 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 t_2 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (t_m <= 1.4e-42) {
		tmp = (2.0 / (t_m * Math.pow(k_m, 2.0))) * ((Math.pow(l, 2.0) * Math.cos(k_m)) / t_2);
	} else if (t_m <= 4.5e+133) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k_m))) * (t_m / (l / Math.tan(k_m))))) / Math.pow((k_m / t_m), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.cos(k_m) / t_m) / t_2)) / Math.pow(k_m, 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):
	t_2 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if t_m <= 1.4e-42:
		tmp = (2.0 / (t_m * math.pow(k_m, 2.0))) * ((math.pow(l, 2.0) * math.cos(k_m)) / t_2)
	elif t_m <= 4.5e+133:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k_m))) * (t_m / (l / math.tan(k_m))))) / math.pow((k_m / t_m), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.cos(k_m) / t_m) / t_2)) / math.pow(k_m, 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)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.4e-42)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k_m ^ 2.0))) * Float64(Float64((l ^ 2.0) * cos(k_m)) / t_2));
	elseif (t_m <= 4.5e+133)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k_m))) * Float64(t_m / Float64(l / tan(k_m))))) / (Float64(k_m / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / t_2)) / (k_m ^ 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)
	t_2 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.4e-42)
		tmp = (2.0 / (t_m * (k_m ^ 2.0))) * (((l ^ 2.0) * cos(k_m)) / t_2);
	elseif (t_m <= 4.5e+133)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / ((k_m / t_m) ^ 2.0);
	else
		tmp = 2.0 * (((l ^ 2.0) * ((cos(k_m) / t_m) / t_2)) / (k_m ^ 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_] := Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.4e-42], N[(N[(2.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+133], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.39999999999999999e-42

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. associate-/r* 32.1%

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

      3. sub-neg 32.1%

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

      4. distribute-rgt-in 27.0%

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

      5. unpow2 27.0%

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

      6. times-frac 19.3%

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

      7. sqr-neg 19.3%

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

      8. times-frac 27.0%

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

      9. unpow2 27.0%

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

      10. distribute-rgt-in 32.1%

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

      11. +-commutative 32.1%

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

      12. associate-+l+ 38.5%

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

   3. Simplified 38.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 78.0%

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

   7. Simplified 78.0%

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

      1. times-frac 78.1%

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

      2. *-commutative 78.1%

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

   9. Applied egg-rr 78.1%

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

   if 1.39999999999999999e-42 < t < 4.49999999999999985e133

   1. Initial program 58.4%

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

      1. associate-/r* 58.3%

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

      2. associate-*l* 58.2%

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

      3. associate-*l/ 58.3%

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

      4. associate-/l* 58.3%

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

      5. +-commutative 58.3%

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

      6. unpow2 58.3%

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

      7. sqr-neg 58.3%

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

      8. distribute-frac-neg 58.3%

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

      9. distribute-frac-neg 58.3%

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

      10. unpow2 58.3%

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

      11. associate--l+ 66.2%

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

      12. metadata-eval 66.2%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 66.2%

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

      14. unpow2 66.2%

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

      15. distribute-frac-neg 66.2%

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

      16. distribute-frac-neg 66.2%

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

      17. sqr-neg 66.2%

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

      18. unpow2 66.2%

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

   3. Simplified 66.2%

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

      1. unpow3 66.2%

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

      2. times-frac 70.2%

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

      3. times-frac 84.6%

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

      4. pow2 84.6%

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

   6. Applied egg-rr 84.6%

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

   if 4.49999999999999985e133 < t

   1. Initial program 8.3%

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

   3. Taylor expanded in t around 0 83.5%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in k around inf 83.5%

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

      1. times-frac 81.2%

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

   8. Simplified 81.2%

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

      1. associate-*l/ 83.8%

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

      2. associate-/r* 83.9%

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

   10. Applied egg-rr 83.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.72 \cdot 10^{-43}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k_m\right)}{t_2 \cdot \left(t_m \cdot {k_m}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 3.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t_m}^{2}}{\frac{\ell}{\sin k_m}} \cdot \frac{t_m}{\frac{\ell}{\tan k_m}}}}{{\left(\frac{k_m}{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{t_2}}{{k_m}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (sin k_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.72e-43)
      (/ (* 2.0 (* (pow l 2.0) (cos k_m))) (* t_2 (* t_m (pow k_m 2.0))))
      (if (<= t_m 3.1e+133)
        (/
         (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k_m))) (/ t_m (/ l (tan k_m)))))
         (pow (/ k_m t_m) 2.0))
        (*
         2.0
         (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) t_2)) (pow k_m 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 t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (t_m <= 1.72e-43) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k_m))) / (t_2 * (t_m * pow(k_m, 2.0)));
	} else if (t_m <= 3.1e+133) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / pow((k_m / t_m), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((cos(k_m) / t_m) / t_2)) / pow(k_m, 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) :: t_2
    real(8) :: tmp
    t_2 = sin(k_m) ** 2.0d0
    if (t_m <= 1.72d-43) then
        tmp = (2.0d0 * ((l ** 2.0d0) * cos(k_m))) / (t_2 * (t_m * (k_m ** 2.0d0)))
    else if (t_m <= 3.1d+133) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / ((k_m / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / t_2)) / (k_m ** 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 t_2 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (t_m <= 1.72e-43) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k_m))) / (t_2 * (t_m * Math.pow(k_m, 2.0)));
	} else if (t_m <= 3.1e+133) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k_m))) * (t_m / (l / Math.tan(k_m))))) / Math.pow((k_m / t_m), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.cos(k_m) / t_m) / t_2)) / Math.pow(k_m, 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):
	t_2 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if t_m <= 1.72e-43:
		tmp = (2.0 * (math.pow(l, 2.0) * math.cos(k_m))) / (t_2 * (t_m * math.pow(k_m, 2.0)))
	elif t_m <= 3.1e+133:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k_m))) * (t_m / (l / math.tan(k_m))))) / math.pow((k_m / t_m), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.cos(k_m) / t_m) / t_2)) / math.pow(k_m, 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)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.72e-43)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k_m))) / Float64(t_2 * Float64(t_m * (k_m ^ 2.0))));
	elseif (t_m <= 3.1e+133)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k_m))) * Float64(t_m / Float64(l / tan(k_m))))) / (Float64(k_m / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / t_2)) / (k_m ^ 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)
	t_2 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.72e-43)
		tmp = (2.0 * ((l ^ 2.0) * cos(k_m))) / (t_2 * (t_m * (k_m ^ 2.0)));
	elseif (t_m <= 3.1e+133)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / ((k_m / t_m) ^ 2.0);
	else
		tmp = 2.0 * (((l ^ 2.0) * ((cos(k_m) / t_m) / t_2)) / (k_m ^ 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_] := Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.72e-43], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.1e+133], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.72000000000000005e-43

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. associate-/r* 32.1%

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

      3. sub-neg 32.1%

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

      4. distribute-rgt-in 27.0%

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

      5. unpow2 27.0%

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

      6. times-frac 19.3%

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

      7. sqr-neg 19.3%

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

      8. times-frac 27.0%

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

      9. unpow2 27.0%

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

      10. distribute-rgt-in 32.1%

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

      11. +-commutative 32.1%

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

      12. associate-+l+ 38.5%

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

   3. Simplified 38.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 78.0%

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

   7. Simplified 78.0%

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

   if 1.72000000000000005e-43 < t < 3.1e133

   1. Initial program 58.4%

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

      1. associate-/r* 58.3%

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

      2. associate-*l* 58.2%

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

      3. associate-*l/ 58.3%

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

      4. associate-/l* 58.3%

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

      5. +-commutative 58.3%

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

      6. unpow2 58.3%

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

      7. sqr-neg 58.3%

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

      8. distribute-frac-neg 58.3%

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

      9. distribute-frac-neg 58.3%

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

      10. unpow2 58.3%

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

      11. associate--l+ 66.2%

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

      12. metadata-eval 66.2%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 66.2%

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

      14. unpow2 66.2%

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

      15. distribute-frac-neg 66.2%

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

      16. distribute-frac-neg 66.2%

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

      17. sqr-neg 66.2%

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

      18. unpow2 66.2%

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

   3. Simplified 66.2%

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

      1. unpow3 66.2%

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

      2. times-frac 70.2%

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

      3. times-frac 84.6%

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

      4. pow2 84.6%

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

   6. Applied egg-rr 84.6%

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

   if 3.1e133 < t

   1. Initial program 8.3%

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

   3. Taylor expanded in t around 0 83.5%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in k around inf 83.5%

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

      1. times-frac 81.2%

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

   8. Simplified 81.2%

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

      1. associate-*l/ 83.8%

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

      2. associate-/r* 83.9%

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

   10. Applied egg-rr 83.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.72 \cdot 10^{-43}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.05 \cdot 10^{-42}:\\ \;\;\;\;\left(2 \cdot \left({\ell}^{2} \cdot \cos k_m\right)\right) \cdot \frac{1}{t_2 \cdot \left(t_m \cdot {k_m}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 2.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t_m}^{2}}{\frac{\ell}{\sin k_m}} \cdot \frac{t_m}{\frac{\ell}{\tan k_m}}}}{{\left(\frac{k_m}{t_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{t_2}}{{k_m}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (sin k_m) 2.0)))
   (*
    t_s
    (if (<= t_m 2.05e-42)
      (*
       (* 2.0 (* (pow l 2.0) (cos k_m)))
       (/ 1.0 (* t_2 (* t_m (pow k_m 2.0)))))
      (if (<= t_m 2.5e+133)
        (/
         (/ 2.0 (* (/ (pow t_m 2.0) (/ l (sin k_m))) (/ t_m (/ l (tan k_m)))))
         (pow (/ k_m t_m) 2.0))
        (*
         2.0
         (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) t_2)) (pow k_m 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 t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (t_m <= 2.05e-42) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k_m))) * (1.0 / (t_2 * (t_m * pow(k_m, 2.0))));
	} else if (t_m <= 2.5e+133) {
		tmp = (2.0 / ((pow(t_m, 2.0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / pow((k_m / t_m), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((cos(k_m) / t_m) / t_2)) / pow(k_m, 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) :: t_2
    real(8) :: tmp
    t_2 = sin(k_m) ** 2.0d0
    if (t_m <= 2.05d-42) then
        tmp = (2.0d0 * ((l ** 2.0d0) * cos(k_m))) * (1.0d0 / (t_2 * (t_m * (k_m ** 2.0d0))))
    else if (t_m <= 2.5d+133) then
        tmp = (2.0d0 / (((t_m ** 2.0d0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / ((k_m / t_m) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / t_2)) / (k_m ** 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 t_2 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (t_m <= 2.05e-42) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k_m))) * (1.0 / (t_2 * (t_m * Math.pow(k_m, 2.0))));
	} else if (t_m <= 2.5e+133) {
		tmp = (2.0 / ((Math.pow(t_m, 2.0) / (l / Math.sin(k_m))) * (t_m / (l / Math.tan(k_m))))) / Math.pow((k_m / t_m), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.cos(k_m) / t_m) / t_2)) / Math.pow(k_m, 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):
	t_2 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if t_m <= 2.05e-42:
		tmp = (2.0 * (math.pow(l, 2.0) * math.cos(k_m))) * (1.0 / (t_2 * (t_m * math.pow(k_m, 2.0))))
	elif t_m <= 2.5e+133:
		tmp = (2.0 / ((math.pow(t_m, 2.0) / (l / math.sin(k_m))) * (t_m / (l / math.tan(k_m))))) / math.pow((k_m / t_m), 2.0)
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.cos(k_m) / t_m) / t_2)) / math.pow(k_m, 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)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 2.05e-42)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k_m))) * Float64(1.0 / Float64(t_2 * Float64(t_m * (k_m ^ 2.0)))));
	elseif (t_m <= 2.5e+133)
		tmp = Float64(Float64(2.0 / Float64(Float64((t_m ^ 2.0) / Float64(l / sin(k_m))) * Float64(t_m / Float64(l / tan(k_m))))) / (Float64(k_m / t_m) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / t_2)) / (k_m ^ 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)
	t_2 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 2.05e-42)
		tmp = (2.0 * ((l ^ 2.0) * cos(k_m))) * (1.0 / (t_2 * (t_m * (k_m ^ 2.0))));
	elseif (t_m <= 2.5e+133)
		tmp = (2.0 / (((t_m ^ 2.0) / (l / sin(k_m))) * (t_m / (l / tan(k_m))))) / ((k_m / t_m) ^ 2.0);
	else
		tmp = 2.0 * (((l ^ 2.0) * ((cos(k_m) / t_m) / t_2)) / (k_m ^ 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_] := Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-42], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$2 * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+133], N[(N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.0500000000000001e-42

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

      2. associate-/r* 32.1%

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

      3. sub-neg 32.1%

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

      4. distribute-rgt-in 27.0%

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

      5. unpow2 27.0%

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

      6. times-frac 19.3%

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

      7. sqr-neg 19.3%

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

      8. times-frac 27.0%

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

      9. unpow2 27.0%

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

      10. distribute-rgt-in 32.1%

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

      11. +-commutative 32.1%

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

      12. associate-+l+ 38.5%

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

   3. Simplified 38.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 78.0%

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

      1. associate-*r/ 78.0%

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

      2. associate-*r* 78.0%

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

   7. Simplified 78.0%

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

      1. div-inv 78.0%

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

      2. *-commutative 78.0%

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

      3. *-commutative 78.0%

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

   9. Applied egg-rr 78.0%

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

   if 2.0500000000000001e-42 < t < 2.4999999999999998e133

   1. Initial program 58.4%

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

      1. associate-/r* 58.3%

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

      2. associate-*l* 58.2%

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

      3. associate-*l/ 58.3%

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

      4. associate-/l* 58.3%

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

      5. +-commutative 58.3%

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

      6. unpow2 58.3%

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

      7. sqr-neg 58.3%

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

      8. distribute-frac-neg 58.3%

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

      9. distribute-frac-neg 58.3%

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

      10. unpow2 58.3%

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

      11. associate--l+ 66.2%

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

      12. metadata-eval 66.2%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k \cdot \tan k}}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 66.2%

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

      14. unpow2 66.2%

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

      15. distribute-frac-neg 66.2%

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

      16. distribute-frac-neg 66.2%

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

      17. sqr-neg 66.2%

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

      18. unpow2 66.2%

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

   3. Simplified 66.2%

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

      1. unpow3 66.2%

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

      2. times-frac 70.2%

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

      3. times-frac 84.6%

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

      4. pow2 84.6%

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

   6. Applied egg-rr 84.6%

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

   if 2.4999999999999998e133 < t

   1. Initial program 8.3%

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

   3. Taylor expanded in t around 0 83.5%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in k around inf 83.5%

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

      1. times-frac 81.2%

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

   8. Simplified 81.2%

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

      1. associate-*l/ 83.8%

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

      2. associate-/r* 83.9%

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

   10. Applied egg-rr 83.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-42}:\\ \;\;\;\;\left(2 \cdot \left({\ell}^{2} \cdot \cos k\right)\right) \cdot \frac{1}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{2}{\frac{{t}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (*
   2.0
   (*
    (/ (pow l 2.0) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m))))
    (pow k_m -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(l, 2.0) / (t_m * (pow(sin(k_m), 2.0) / cos(k_m)))) * pow(k_m, -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 * (((l ** 2.0d0) / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m)))) * (k_m ** (-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(l, 2.0) / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)))) * Math.pow(k_m, -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(l, 2.0) / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))) * math.pow(k_m, -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(Float64((l ^ 2.0) / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m)))) * (k_m ^ -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 * (((l ^ 2.0) / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m)))) * (k_m ^ -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[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Taylor expanded in t around 0 78.3%

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

   1. associate-/l* 79.2%

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

5. Simplified 79.2%

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

6. Taylor expanded in k around inf 78.3%

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

   1. times-frac 78.7%

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

8. Simplified 78.7%

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

   1. associate-*l/ 79.1%

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

   2. associate-/r* 79.2%

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

10. Applied egg-rr 79.2%

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

    1. expm1-log1p-u 53.8%

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

    2. expm1-udef 46.9%

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

    3. div-inv 46.9%

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

    4. associate-*r/ 46.6%

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

    5. pow-flip 46.6%

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

    6. metadata-eval 46.6%

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

12. Applied egg-rr 46.6%

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

    1. expm1-def 53.5%

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

    2. expm1-log1p 76.0%

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

    3. associate-/l* 79.0%

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

    4. associate-/r/ 79.1%

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

14. Simplified 79.1%

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

15. Final simplification 79.1%

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

Alternative 8: 73.5% accurate, 0.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (*
   2.0
   (/
    (* (pow l 2.0) (/ (/ (cos k_m) t_m) (pow (sin k_m) 2.0)))
    (pow k_m 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(l, 2.0) * ((cos(k_m) / t_m) / pow(sin(k_m), 2.0))) / pow(k_m, 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 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / (sin(k_m) ** 2.0d0))) / (k_m ** 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(l, 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(Math.sin(k_m), 2.0))) / Math.pow(k_m, 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(l, 2.0) * ((math.cos(k_m) / t_m) / math.pow(math.sin(k_m), 2.0))) / math.pow(k_m, 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(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (sin(k_m) ^ 2.0))) / (k_m ^ 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 * (((l ^ 2.0) * ((cos(k_m) / t_m) / (sin(k_m) ^ 2.0))) / (k_m ^ 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[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Taylor expanded in t around 0 78.3%

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

   1. associate-/l* 79.2%

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

5. Simplified 79.2%

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

6. Taylor expanded in k around inf 78.3%

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

   1. times-frac 78.7%

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

8. Simplified 78.7%

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

   1. associate-*l/ 79.1%

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

   2. associate-/r* 79.2%

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

10. Applied egg-rr 79.2%

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

11. Final simplification 79.2%

    \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}}{{k}^{2}} \]
12. Add Preprocessing

Alternative 9: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{{k_m}^{2}}}{{k_m}^{2}}\\ \mathbf{elif}\;t_m \leq 2.15 \cdot 10^{+133}:\\ \;\;\;\;\frac{2}{k_m} \cdot \left({t_m}^{-2} \cdot \frac{t_m \cdot {\ell}^{2}}{\tan k_m \cdot \left(k_m \cdot \sin k_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t_m \cdot {k_m}^{2}} - \frac{0.16666666666666666}{t_m}\right)}{{k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-155)
    (*
     2.0
     (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) (pow k_m 2.0))) (pow k_m 2.0)))
    (if (<= t_m 2.15e+133)
      (*
       (/ 2.0 k_m)
       (*
        (pow t_m -2.0)
        (/ (* t_m (pow l 2.0)) (* (tan k_m) (* k_m (sin k_m))))))
      (*
       2.0
       (/
        (*
         (pow l 2.0)
         (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 t_m)))
        (pow k_m 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 (t_m <= 8e-155) {
		tmp = 2.0 * ((pow(l, 2.0) * ((cos(k_m) / t_m) / pow(k_m, 2.0))) / pow(k_m, 2.0));
	} else if (t_m <= 2.15e+133) {
		tmp = (2.0 / k_m) * (pow(t_m, -2.0) * ((t_m * pow(l, 2.0)) / (tan(k_m) * (k_m * sin(k_m)))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / pow(k_m, 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 (t_m <= 8d-155) then
        tmp = 2.0d0 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / (k_m ** 2.0d0))) / (k_m ** 2.0d0))
    else if (t_m <= 2.15d+133) then
        tmp = (2.0d0 / k_m) * ((t_m ** (-2.0d0)) * ((t_m * (l ** 2.0d0)) / (tan(k_m) * (k_m * sin(k_m)))))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / t_m))) / (k_m ** 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 (t_m <= 8e-155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(k_m, 2.0))) / Math.pow(k_m, 2.0));
	} else if (t_m <= 2.15e+133) {
		tmp = (2.0 / k_m) * (Math.pow(t_m, -2.0) * ((t_m * Math.pow(l, 2.0)) / (Math.tan(k_m) * (k_m * Math.sin(k_m)))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / Math.pow(k_m, 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 t_m <= 8e-155:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.cos(k_m) / t_m) / math.pow(k_m, 2.0))) / math.pow(k_m, 2.0))
	elif t_m <= 2.15e+133:
		tmp = (2.0 / k_m) * (math.pow(t_m, -2.0) * ((t_m * math.pow(l, 2.0)) / (math.tan(k_m) * (k_m * math.sin(k_m)))))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / math.pow(k_m, 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 (t_m <= 8e-155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 2.0)));
	elseif (t_m <= 2.15e+133)
		tmp = Float64(Float64(2.0 / k_m) * Float64((t_m ^ -2.0) * Float64(Float64(t_m * (l ^ 2.0)) / Float64(tan(k_m) * Float64(k_m * sin(k_m))))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / t_m))) / (k_m ^ 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 (t_m <= 8e-155)
		tmp = 2.0 * (((l ^ 2.0) * ((cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 2.0));
	elseif (t_m <= 2.15e+133)
		tmp = (2.0 / k_m) * ((t_m ^ -2.0) * ((t_m * (l ^ 2.0)) / (tan(k_m) * (k_m * sin(k_m)))));
	else
		tmp = 2.0 * (((l ^ 2.0) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / t_m))) / (k_m ^ 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[t$95$m, 8e-155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.15e+133], N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(N[Power[t$95$m, -2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.00000000000000011e-155

   1. Initial program 29.0%

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

   3. Taylor expanded in t around 0 77.2%

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

      1. associate-/l* 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in k around inf 77.2%

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

      1. times-frac 77.9%

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

   8. Simplified 77.9%

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

      1. associate-*l/ 78.3%

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

      2. associate-/r* 78.4%

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

   10. Applied egg-rr 78.4%

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

   11. Taylor expanded in k around 0 71.3%

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

   if 8.00000000000000011e-155 < t < 2.14999999999999997e133

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. associate-/r* 56.5%

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

      3. sub-neg 56.5%

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

      4. distribute-rgt-in 56.3%

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

      5. unpow2 56.3%

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

      6. times-frac 56.3%

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

      7. sqr-neg 56.3%

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

      8. times-frac 56.3%

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

      9. unpow2 56.3%

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

      10. distribute-rgt-in 56.5%

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

      11. +-commutative 56.5%

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

      12. associate-+l+ 60.6%

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

   3. Simplified 60.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 60.7%

         \[\leadsto \color{blue}{\frac{2}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      2. +-rgt-identity 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-*r* 60.7%

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

      5. *-commutative 60.7%

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

      6. associate-*l* 60.6%

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

      7. associate-/r/ 60.6%

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

      8. associate-/l/ 60.6%

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

      9. associate-/r/ 58.6%

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

      10. unpow2 58.6%

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

   6. Applied egg-rr 62.7%

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

   7. Taylor expanded in t around 0 78.5%

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

      1. associate-/r* 78.6%

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

   9. Simplified 78.6%

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

       1. expm1-log1p-u 75.9%

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

       2. expm1-udef 66.5%

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

       3. div-inv 66.5%

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

       4. pow-flip 66.5%

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

       5. metadata-eval 66.5%

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

       6. associate-/r/ 66.5%

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

   11. Applied egg-rr 66.5%

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

       1. expm1-def 77.6%

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

       2. expm1-log1p 80.4%

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

       3. associate-*l* 80.4%

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

       4. associate-/l/ 80.4%

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

       5. associate-*l/ 78.5%

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

       6. associate-*r* 78.5%

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

   13. Simplified 78.5%

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

   if 2.14999999999999997e133 < t

   1. Initial program 8.3%

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

   3. Taylor expanded in t around 0 83.5%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in k around inf 83.5%

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

      1. times-frac 81.2%

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

   8. Simplified 81.2%

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

      1. associate-*l/ 83.8%

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

      2. associate-/r* 83.9%

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

   10. Applied egg-rr 83.9%

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

   11. Taylor expanded in k around 0 81.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}}{{k}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative 81.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{{k}^{2}} \]

       2. associate-*r/ 81.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)}{{k}^{2}} \]

       3. metadata-eval 81.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{\color{blue}{0.16666666666666666}}{t}\right)}{{k}^{2}} \]

   13. Simplified 81.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}}{{k}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+133}:\\ \;\;\;\;\frac{2}{k} \cdot \left({t}^{-2} \cdot \frac{t \cdot {\ell}^{2}}{\tan k \cdot \left(k \cdot \sin k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.36 \cdot 10^{-167}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{{k_m}^{2}}}{{k_m}^{2}}\\ \mathbf{elif}\;t_m \leq 8 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{\sin k_m \cdot \tan k_m} \cdot \frac{2}{k_m}}{\frac{k_m}{t_m} \cdot {t_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t_m \cdot {k_m}^{2}} - \frac{0.16666666666666666}{t_m}\right)}{{k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.36e-167)
    (*
     2.0
     (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) (pow k_m 2.0))) (pow k_m 2.0)))
    (if (<= t_m 8e+125)
      (/
       (* (/ (pow l 2.0) (* (sin k_m) (tan k_m))) (/ 2.0 k_m))
       (* (/ k_m t_m) (pow t_m 2.0)))
      (*
       2.0
       (/
        (*
         (pow l 2.0)
         (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 t_m)))
        (pow k_m 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 (t_m <= 1.36e-167) {
		tmp = 2.0 * ((pow(l, 2.0) * ((cos(k_m) / t_m) / pow(k_m, 2.0))) / pow(k_m, 2.0));
	} else if (t_m <= 8e+125) {
		tmp = ((pow(l, 2.0) / (sin(k_m) * tan(k_m))) * (2.0 / k_m)) / ((k_m / t_m) * pow(t_m, 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / pow(k_m, 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 (t_m <= 1.36d-167) then
        tmp = 2.0d0 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / (k_m ** 2.0d0))) / (k_m ** 2.0d0))
    else if (t_m <= 8d+125) then
        tmp = (((l ** 2.0d0) / (sin(k_m) * tan(k_m))) * (2.0d0 / k_m)) / ((k_m / t_m) * (t_m ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / t_m))) / (k_m ** 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 (t_m <= 1.36e-167) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(k_m, 2.0))) / Math.pow(k_m, 2.0));
	} else if (t_m <= 8e+125) {
		tmp = ((Math.pow(l, 2.0) / (Math.sin(k_m) * Math.tan(k_m))) * (2.0 / k_m)) / ((k_m / t_m) * Math.pow(t_m, 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / Math.pow(k_m, 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 t_m <= 1.36e-167:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.cos(k_m) / t_m) / math.pow(k_m, 2.0))) / math.pow(k_m, 2.0))
	elif t_m <= 8e+125:
		tmp = ((math.pow(l, 2.0) / (math.sin(k_m) * math.tan(k_m))) * (2.0 / k_m)) / ((k_m / t_m) * math.pow(t_m, 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / math.pow(k_m, 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 (t_m <= 1.36e-167)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 2.0)));
	elseif (t_m <= 8e+125)
		tmp = Float64(Float64(Float64((l ^ 2.0) / Float64(sin(k_m) * tan(k_m))) * Float64(2.0 / k_m)) / Float64(Float64(k_m / t_m) * (t_m ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / t_m))) / (k_m ^ 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 (t_m <= 1.36e-167)
		tmp = 2.0 * (((l ^ 2.0) * ((cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 2.0));
	elseif (t_m <= 8e+125)
		tmp = (((l ^ 2.0) / (sin(k_m) * tan(k_m))) * (2.0 / k_m)) / ((k_m / t_m) * (t_m ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / t_m))) / (k_m ^ 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[t$95$m, 1.36e-167], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+125], N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.36000000000000009e-167

   1. Initial program 29.0%

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

   3. Taylor expanded in t around 0 77.3%

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

      1. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in k around inf 77.2%

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

      1. times-frac 78.6%

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

   8. Simplified 78.6%

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

      1. associate-*l/ 78.4%

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

      2. associate-/r* 78.5%

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

   10. Applied egg-rr 78.5%

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

   11. Taylor expanded in k around 0 71.8%

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

   if 1.36000000000000009e-167 < t < 7.9999999999999994e125

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. associate-/r* 55.3%

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

      3. sub-neg 55.3%

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

      4. distribute-rgt-in 55.1%

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

      5. unpow2 55.1%

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

      6. times-frac 55.1%

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

      7. sqr-neg 55.1%

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

      8. times-frac 55.1%

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

      9. unpow2 55.1%

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

      10. distribute-rgt-in 55.3%

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

      11. +-commutative 55.3%

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

      12. associate-+l+ 57.3%

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

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l/ 57.4%

         \[\leadsto \color{blue}{\frac{2}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      2. +-rgt-identity 57.4%

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

      3. *-commutative 57.4%

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

      4. associate-*r* 57.3%

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

      5. *-commutative 57.3%

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

      6. associate-*l* 57.3%

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

      7. associate-/r/ 57.3%

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

      8. associate-/l/ 57.3%

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

      9. associate-/r/ 55.4%

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

      10. unpow2 55.4%

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

   6. Applied egg-rr 59.3%

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

   7. Taylor expanded in t around 0 74.3%

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

      1. associate-/r* 74.3%

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

   9. Simplified 74.3%

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

       1. frac-times 77.8%

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

   11. Applied egg-rr 77.8%

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

   if 7.9999999999999994e125 < t

   1. Initial program 8.1%

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

   3. Taylor expanded in t around 0 84.0%

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

      1. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

   6. Taylor expanded in k around inf 84.0%

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

      1. times-frac 81.7%

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

   8. Simplified 81.7%

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

      1. associate-*l/ 84.3%

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

      2. associate-/r* 84.3%

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

   10. Applied egg-rr 84.3%

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

   11. Taylor expanded in k around 0 82.0%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}}{{k}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative 82.0%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{{k}^{2}} \]

       2. associate-*r/ 82.0%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)}{{k}^{2}} \]

       3. metadata-eval 82.0%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{\color{blue}{0.16666666666666666}}{t}\right)}{{k}^{2}} \]

   13. Simplified 82.0%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}}{{k}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.36 \cdot 10^{-167}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}}{{k}^{2}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{\sin k \cdot \tan k} \cdot \frac{2}{k}}{\frac{k}{t} \cdot {t}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k_m}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 7.6 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{t_2}{{k_m}^{2}}}{{k_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{t_2}{0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}}}{{k_m}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 7.6e-5)
      (* 2.0 (/ (* (pow l 2.0) (/ t_2 (pow k_m 2.0))) (pow k_m 2.0)))
      (*
       2.0
       (/
        (* (pow l 2.0) (/ t_2 (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
        (pow k_m 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 t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 7.6e-5) {
		tmp = 2.0 * ((pow(l, 2.0) * (t_2 / pow(k_m, 2.0))) / pow(k_m, 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * (t_2 / (0.5 - (cos((k_m * 2.0)) / 2.0)))) / pow(k_m, 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) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 7.6d-5) then
        tmp = 2.0d0 * (((l ** 2.0d0) * (t_2 / (k_m ** 2.0d0))) / (k_m ** 2.0d0))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * (t_2 / (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))) / (k_m ** 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 t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 7.6e-5) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * (t_2 / Math.pow(k_m, 2.0))) / Math.pow(k_m, 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * (t_2 / (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / Math.pow(k_m, 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):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 7.6e-5:
		tmp = 2.0 * ((math.pow(l, 2.0) * (t_2 / math.pow(k_m, 2.0))) / math.pow(k_m, 2.0))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * (t_2 / (0.5 - (math.cos((k_m * 2.0)) / 2.0)))) / math.pow(k_m, 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)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 7.6e-5)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(t_2 / (k_m ^ 2.0))) / (k_m ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(t_2 / Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / (k_m ^ 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)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 7.6e-5)
		tmp = 2.0 * (((l ^ 2.0) * (t_2 / (k_m ^ 2.0))) / (k_m ^ 2.0));
	else
		tmp = 2.0 * (((l ^ 2.0) * (t_2 / (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (k_m ^ 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_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 7.6e-5], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(t$95$2 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(t$95$2 / N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.6000000000000004e-5

   1. Initial program 34.0%

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

   3. Taylor expanded in t around 0 81.2%

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

      1. associate-/l* 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in k around inf 81.2%

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

      1. times-frac 80.9%

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

   8. Simplified 80.9%

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

      1. associate-*l/ 82.3%

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

      2. associate-/r* 82.3%

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

   10. Applied egg-rr 82.3%

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

   11. Taylor expanded in k around 0 77.5%

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

   if 7.6000000000000004e-5 < k

   1. Initial program 24.8%

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

   3. Taylor expanded in t around 0 70.6%

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

      1. associate-/l* 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in k around inf 70.5%

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

      1. times-frac 72.9%

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

   8. Simplified 72.9%

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

      1. associate-*l/ 70.7%

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

      2. associate-/r* 70.8%

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

   10. Applied egg-rr 70.8%

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

       1. unpow2 70.8%

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

       2. sin-mult 70.7%

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

   12. Applied egg-rr 70.7%

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

       1. div-sub 70.7%

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

       2. +-inverses 70.7%

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

       3. cos-0 70.7%

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

       4. metadata-eval 70.7%

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

       5. count-2 70.7%

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

   14. Simplified 70.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 7.6 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k_m}{t_m}}{{k_m}^{2}}}{{k_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\frac{{\ell}^{2} \cdot \cos k_m}{t_m \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.6e-5)
    (*
     2.0
     (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) (pow k_m 2.0))) (pow k_m 2.0)))
    (/
     2.0
     (/
      (pow k_m 2.0)
      (/
       (* (pow l 2.0) (cos k_m))
       (* t_m (- 0.5 (/ (cos (* k_m 2.0)) 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 <= 7.6e-5) {
		tmp = 2.0 * ((pow(l, 2.0) * ((cos(k_m) / t_m) / pow(k_m, 2.0))) / pow(k_m, 2.0));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) / ((pow(l, 2.0) * cos(k_m)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 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 <= 7.6d-5) then
        tmp = 2.0d0 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / (k_m ** 2.0d0))) / (k_m ** 2.0d0))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) / (((l ** 2.0d0) * cos(k_m)) / (t_m * (0.5d0 - (cos((k_m * 2.0d0)) / 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 <= 7.6e-5) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(k_m, 2.0))) / Math.pow(k_m, 2.0));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) / ((Math.pow(l, 2.0) * Math.cos(k_m)) / (t_m * (0.5 - (Math.cos((k_m * 2.0)) / 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 <= 7.6e-5:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((math.cos(k_m) / t_m) / math.pow(k_m, 2.0))) / math.pow(k_m, 2.0))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) / ((math.pow(l, 2.0) * math.cos(k_m)) / (t_m * (0.5 - (math.cos((k_m * 2.0)) / 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 <= 7.6e-5)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) / Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64(t_m * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 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 <= 7.6e-5)
		tmp = 2.0 * (((l ^ 2.0) * ((cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 2.0));
	else
		tmp = 2.0 / ((k_m ^ 2.0) / (((l ^ 2.0) * cos(k_m)) / (t_m * (0.5 - (cos((k_m * 2.0)) / 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, 7.6e-5], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.6000000000000004e-5

   1. Initial program 34.0%

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

   3. Taylor expanded in t around 0 81.2%

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

      1. associate-/l* 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in k around inf 81.2%

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

      1. times-frac 80.9%

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

   8. Simplified 80.9%

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

      1. associate-*l/ 82.3%

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

      2. associate-/r* 82.3%

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

   10. Applied egg-rr 82.3%

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

   11. Taylor expanded in k around 0 77.5%

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

   if 7.6000000000000004e-5 < k

   1. Initial program 24.8%

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

   3. Taylor expanded in t around 0 70.6%

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

      1. associate-/l* 70.8%

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

   5. Simplified 70.8%

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

      1. unpow2 70.8%

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

      2. sin-mult 70.7%

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

   7. Applied egg-rr 70.7%

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

      1. div-sub 70.7%

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

      2. +-inverses 70.7%

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

      3. cos-0 70.7%

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

      4. metadata-eval 70.7%

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

      5. count-2 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\frac{{\ell}^{2} \cdot \cos k}{t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.6% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (*
   2.0
   (/ (* (pow l 2.0) (/ (/ (cos k_m) t_m) (pow k_m 2.0))) (pow k_m 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(l, 2.0) * ((cos(k_m) / t_m) / pow(k_m, 2.0))) / pow(k_m, 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 * (((l ** 2.0d0) * ((cos(k_m) / t_m) / (k_m ** 2.0d0))) / (k_m ** 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(l, 2.0) * ((Math.cos(k_m) / t_m) / Math.pow(k_m, 2.0))) / Math.pow(k_m, 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(l, 2.0) * ((math.cos(k_m) / t_m) / math.pow(k_m, 2.0))) / math.pow(k_m, 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(Float64((l ^ 2.0) * Float64(Float64(cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 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 * (((l ^ 2.0) * ((cos(k_m) / t_m) / (k_m ^ 2.0))) / (k_m ^ 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[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Taylor expanded in t around 0 78.3%

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

   1. associate-/l* 79.2%

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

5. Simplified 79.2%

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

6. Taylor expanded in k around inf 78.3%

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

   1. times-frac 78.7%

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

8. Simplified 78.7%

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

   1. associate-*l/ 79.1%

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

   2. associate-/r* 79.2%

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

10. Applied egg-rr 79.2%

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

11. Taylor expanded in k around 0 71.7%

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

12. Final simplification 71.7%

    \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}}{{k}^{2}} \]
13. Add Preprocessing

Alternative 14: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k_m\right)}{t_m \cdot {k_m}^{4}}\\ \mathbf{elif}\;t_m \leq 4.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \left(\sin k_m \cdot \frac{{t_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}}\\ \mathbf{elif}\;t_m \leq 1.3 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(k_m \cdot \frac{\frac{k_m}{t_m} \cdot {t_m}^{1.5}}{\ell}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t_m \cdot {k_m}^{2}} - \frac{0.16666666666666666}{t_m}\right)}{{k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.2e-77)
    (/ (* 2.0 (* (pow l 2.0) (cos k_m))) (* t_m (pow k_m 4.0)))
    (if (<= t_m 4.8e+42)
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t_m 3.0) (* l l))))
        (/ k_m (* t_m (/ t_m k_m)))))
      (if (<= t_m 1.3e+158)
        (sqrt (/ 4.0 (pow (* k_m (/ (* (/ k_m t_m) (pow t_m 1.5)) l)) 4.0)))
        (*
         2.0
         (/
          (*
           (pow l 2.0)
           (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 t_m)))
          (pow k_m 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 (t_m <= 8.2e-77) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k_m))) / (t_m * pow(k_m, 4.0));
	} else if (t_m <= 4.8e+42) {
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * (pow(t_m, 3.0) / (l * l)))) * (k_m / (t_m * (t_m / k_m))));
	} else if (t_m <= 1.3e+158) {
		tmp = sqrt((4.0 / pow((k_m * (((k_m / t_m) * pow(t_m, 1.5)) / l)), 4.0)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / pow(k_m, 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 (t_m <= 8.2d-77) then
        tmp = (2.0d0 * ((l ** 2.0d0) * cos(k_m))) / (t_m * (k_m ** 4.0d0))
    else if (t_m <= 4.8d+42) then
        tmp = 2.0d0 / ((tan(k_m) * (sin(k_m) * ((t_m ** 3.0d0) / (l * l)))) * (k_m / (t_m * (t_m / k_m))))
    else if (t_m <= 1.3d+158) then
        tmp = sqrt((4.0d0 / ((k_m * (((k_m / t_m) * (t_m ** 1.5d0)) / l)) ** 4.0d0)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / t_m))) / (k_m ** 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 (t_m <= 8.2e-77) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k_m))) / (t_m * Math.pow(k_m, 4.0));
	} else if (t_m <= 4.8e+42) {
		tmp = 2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l)))) * (k_m / (t_m * (t_m / k_m))));
	} else if (t_m <= 1.3e+158) {
		tmp = Math.sqrt((4.0 / Math.pow((k_m * (((k_m / t_m) * Math.pow(t_m, 1.5)) / l)), 4.0)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / Math.pow(k_m, 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 t_m <= 8.2e-77:
		tmp = (2.0 * (math.pow(l, 2.0) * math.cos(k_m))) / (t_m * math.pow(k_m, 4.0))
	elif t_m <= 4.8e+42:
		tmp = 2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l)))) * (k_m / (t_m * (t_m / k_m))))
	elif t_m <= 1.3e+158:
		tmp = math.sqrt((4.0 / math.pow((k_m * (((k_m / t_m) * math.pow(t_m, 1.5)) / l)), 4.0)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / math.pow(k_m, 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 (t_m <= 8.2e-77)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k_m))) / Float64(t_m * (k_m ^ 4.0)));
	elseif (t_m <= 4.8e+42)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(k_m / Float64(t_m * Float64(t_m / k_m)))));
	elseif (t_m <= 1.3e+158)
		tmp = sqrt(Float64(4.0 / (Float64(k_m * Float64(Float64(Float64(k_m / t_m) * (t_m ^ 1.5)) / l)) ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / t_m))) / (k_m ^ 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 (t_m <= 8.2e-77)
		tmp = (2.0 * ((l ^ 2.0) * cos(k_m))) / (t_m * (k_m ^ 4.0));
	elseif (t_m <= 4.8e+42)
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * ((t_m ^ 3.0) / (l * l)))) * (k_m / (t_m * (t_m / k_m))));
	elseif (t_m <= 1.3e+158)
		tmp = sqrt((4.0 / ((k_m * (((k_m / t_m) * (t_m ^ 1.5)) / l)) ^ 4.0)));
	else
		tmp = 2.0 * (((l ^ 2.0) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / t_m))) / (k_m ^ 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[t$95$m, 8.2e-77], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+42], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+158], N[Sqrt[N[(4.0 / N[Power[N[(k$95$m * N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 8.19999999999999925e-77

   1. Initial program 29.9%

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

      1. associate-*l* 29.9%

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

      2. associate-/r* 29.9%

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

      3. sub-neg 29.9%

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

      4. distribute-rgt-in 24.5%

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

      5. unpow2 24.5%

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

      6. times-frac 16.4%

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

      7. sqr-neg 16.4%

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

      8. times-frac 24.5%

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

      9. unpow2 24.5%

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

      10. distribute-rgt-in 29.9%

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

      11. +-commutative 29.9%

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

      12. associate-+l+ 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. associate-*r* 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in k around 0 65.6%

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

   if 8.19999999999999925e-77 < t < 4.7999999999999997e42

   1. Initial program 79.5%

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

      1. +-commutative 79.5%

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

      2. associate--l+ 79.5%

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

      3. metadata-eval 79.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)} \]

      4. +-rgt-identity 79.5%

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

      5. unpow2 79.5%

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

      6. clear-num 79.5%

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

      7. frac-times 79.5%

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

      8. *-un-lft-identity 79.5%

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

   4. Applied egg-rr 79.5%

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

   if 4.7999999999999997e42 < t < 1.3e158

   1. Initial program 32.2%

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

      1. associate-*l* 32.2%

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

      2. associate-/r* 32.2%

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

      3. sub-neg 32.2%

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

      4. distribute-rgt-in 31.7%

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

      5. unpow2 31.7%

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

      6. times-frac 31.7%

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

      7. sqr-neg 31.7%

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

      8. times-frac 31.7%

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

      9. unpow2 31.7%

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

      10. distribute-rgt-in 32.2%

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

      11. +-commutative 32.2%

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

      12. associate-+l+ 45.1%

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

   3. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 63.1%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}}} \]

      2. pow-sqr 63.1%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{\left(2 \cdot 2\right)}}}} \]

      3. associate-*r* 63.1%

         \[\leadsto \sqrt{\frac{4}{{\color{blue}{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}}^{\left(2 \cdot 2\right)}}} \]

      4. associate-*r/ 75.4%

         \[\leadsto \sqrt{\frac{4}{{\left(\color{blue}{\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\left(2 \cdot 2\right)}}} \]

      5. metadata-eval 75.4%

         \[\leadsto \sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\color{blue}{4}}}} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{4}}}} \]

   9. Taylor expanded in k around 0 75.9%

      \[\leadsto \sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{4}}} \]

   if 1.3e158 < t

   1. Initial program 3.2%

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

   3. Taylor expanded in t around 0 80.9%

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

      1. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in k around inf 80.9%

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

      1. times-frac 78.2%

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

   8. Simplified 78.2%

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

      1. associate-*l/ 81.2%

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

      2. associate-/r* 81.2%

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

   10. Applied egg-rr 81.2%

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

   11. Taylor expanded in k around 0 78.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}}{{k}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative 78.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{{k}^{2}} \]

       2. associate-*r/ 78.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)}{{k}^{2}} \]

       3. metadata-eval 78.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{\color{blue}{0.16666666666666666}}{t}\right)}{{k}^{2}} \]

   13. Simplified 78.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}}{{k}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \frac{k}{t \cdot \frac{t}{k}}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+158}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(k \cdot \frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.95 \cdot 10^{-80}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k_m\right)}{t_m \cdot {k_m}^{4}}\\ \mathbf{elif}\;t_m \leq 2.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \left(\sin k_m \cdot \frac{{t_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \frac{\frac{k_m}{t_m}}{\frac{t_m}{k_m}}}\\ \mathbf{elif}\;t_m \leq 8 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(k_m \cdot \frac{\frac{k_m}{t_m} \cdot {t_m}^{1.5}}{\ell}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t_m \cdot {k_m}^{2}} - \frac{0.16666666666666666}{t_m}\right)}{{k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-80)
    (/ (* 2.0 (* (pow l 2.0) (cos k_m))) (* t_m (pow k_m 4.0)))
    (if (<= t_m 2.1e+43)
      (/
       2.0
       (*
        (* (tan k_m) (* (sin k_m) (/ (pow t_m 3.0) (* l l))))
        (/ (/ k_m t_m) (/ t_m k_m))))
      (if (<= t_m 8e+156)
        (sqrt (/ 4.0 (pow (* k_m (/ (* (/ k_m t_m) (pow t_m 1.5)) l)) 4.0)))
        (*
         2.0
         (/
          (*
           (pow l 2.0)
           (- (/ 1.0 (* t_m (pow k_m 2.0))) (/ 0.16666666666666666 t_m)))
          (pow k_m 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 (t_m <= 1.95e-80) {
		tmp = (2.0 * (pow(l, 2.0) * cos(k_m))) / (t_m * pow(k_m, 4.0));
	} else if (t_m <= 2.1e+43) {
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * (pow(t_m, 3.0) / (l * l)))) * ((k_m / t_m) / (t_m / k_m)));
	} else if (t_m <= 8e+156) {
		tmp = sqrt((4.0 / pow((k_m * (((k_m / t_m) * pow(t_m, 1.5)) / l)), 4.0)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * ((1.0 / (t_m * pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / pow(k_m, 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 (t_m <= 1.95d-80) then
        tmp = (2.0d0 * ((l ** 2.0d0) * cos(k_m))) / (t_m * (k_m ** 4.0d0))
    else if (t_m <= 2.1d+43) then
        tmp = 2.0d0 / ((tan(k_m) * (sin(k_m) * ((t_m ** 3.0d0) / (l * l)))) * ((k_m / t_m) / (t_m / k_m)))
    else if (t_m <= 8d+156) then
        tmp = sqrt((4.0d0 / ((k_m * (((k_m / t_m) * (t_m ** 1.5d0)) / l)) ** 4.0d0)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) * ((1.0d0 / (t_m * (k_m ** 2.0d0))) - (0.16666666666666666d0 / t_m))) / (k_m ** 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 (t_m <= 1.95e-80) {
		tmp = (2.0 * (Math.pow(l, 2.0) * Math.cos(k_m))) / (t_m * Math.pow(k_m, 4.0));
	} else if (t_m <= 2.1e+43) {
		tmp = 2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l)))) * ((k_m / t_m) / (t_m / k_m)));
	} else if (t_m <= 8e+156) {
		tmp = Math.sqrt((4.0 / Math.pow((k_m * (((k_m / t_m) * Math.pow(t_m, 1.5)) / l)), 4.0)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * ((1.0 / (t_m * Math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / Math.pow(k_m, 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 t_m <= 1.95e-80:
		tmp = (2.0 * (math.pow(l, 2.0) * math.cos(k_m))) / (t_m * math.pow(k_m, 4.0))
	elif t_m <= 2.1e+43:
		tmp = 2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l)))) * ((k_m / t_m) / (t_m / k_m)))
	elif t_m <= 8e+156:
		tmp = math.sqrt((4.0 / math.pow((k_m * (((k_m / t_m) * math.pow(t_m, 1.5)) / l)), 4.0)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) * ((1.0 / (t_m * math.pow(k_m, 2.0))) - (0.16666666666666666 / t_m))) / math.pow(k_m, 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 (t_m <= 1.95e-80)
		tmp = Float64(Float64(2.0 * Float64((l ^ 2.0) * cos(k_m))) / Float64(t_m * (k_m ^ 4.0)));
	elseif (t_m <= 2.1e+43)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(Float64(k_m / t_m) / Float64(t_m / k_m))));
	elseif (t_m <= 8e+156)
		tmp = sqrt(Float64(4.0 / (Float64(k_m * Float64(Float64(Float64(k_m / t_m) * (t_m ^ 1.5)) / l)) ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(Float64(1.0 / Float64(t_m * (k_m ^ 2.0))) - Float64(0.16666666666666666 / t_m))) / (k_m ^ 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 (t_m <= 1.95e-80)
		tmp = (2.0 * ((l ^ 2.0) * cos(k_m))) / (t_m * (k_m ^ 4.0));
	elseif (t_m <= 2.1e+43)
		tmp = 2.0 / ((tan(k_m) * (sin(k_m) * ((t_m ^ 3.0) / (l * l)))) * ((k_m / t_m) / (t_m / k_m)));
	elseif (t_m <= 8e+156)
		tmp = sqrt((4.0 / ((k_m * (((k_m / t_m) * (t_m ^ 1.5)) / l)) ^ 4.0)));
	else
		tmp = 2.0 * (((l ^ 2.0) * ((1.0 / (t_m * (k_m ^ 2.0))) - (0.16666666666666666 / t_m))) / (k_m ^ 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[t$95$m, 1.95e-80], N[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+43], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] / N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+156], N[Sqrt[N[(4.0 / N[Power[N[(k$95$m * N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 1.9499999999999999e-80

   1. Initial program 29.9%

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

      1. associate-*l* 29.9%

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

      2. associate-/r* 29.9%

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

      3. sub-neg 29.9%

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

      4. distribute-rgt-in 24.5%

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

      5. unpow2 24.5%

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

      6. times-frac 16.4%

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

      7. sqr-neg 16.4%

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

      8. times-frac 24.5%

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

      9. unpow2 24.5%

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

      10. distribute-rgt-in 29.9%

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

      11. +-commutative 29.9%

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

      12. associate-+l+ 36.5%

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

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. associate-*r* 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in k around 0 65.6%

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

   if 1.9499999999999999e-80 < t < 2.10000000000000002e43

   1. Initial program 79.5%

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

      1. +-commutative 79.5%

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

      2. associate--l+ 79.5%

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

      3. metadata-eval 79.5%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)} \]

      4. +-rgt-identity 79.5%

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

      5. unpow2 79.5%

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

      6. clear-num 79.5%

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

      7. un-div-inv 79.4%

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

   4. Applied egg-rr 79.4%

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

   if 2.10000000000000002e43 < t < 7.9999999999999999e156

   1. Initial program 32.2%

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

      1. associate-*l* 32.2%

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

      2. associate-/r* 32.2%

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

      3. sub-neg 32.2%

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

      4. distribute-rgt-in 31.7%

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

      5. unpow2 31.7%

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

      6. times-frac 31.7%

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

      7. sqr-neg 31.7%

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

      8. times-frac 31.7%

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

      9. unpow2 31.7%

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

      10. distribute-rgt-in 32.2%

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

      11. +-commutative 32.2%

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

      12. associate-+l+ 45.1%

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

   3. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   6. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left({\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}\right)}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 63.1%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}}} \]

      2. pow-sqr 63.1%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{\left(\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{\left(2 \cdot 2\right)}}}} \]

      3. associate-*r* 63.1%

         \[\leadsto \sqrt{\frac{4}{{\color{blue}{\left(\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \sqrt{\sin k \cdot \tan k}\right)}}^{\left(2 \cdot 2\right)}}} \]

      4. associate-*r/ 75.4%

         \[\leadsto \sqrt{\frac{4}{{\left(\color{blue}{\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\left(2 \cdot 2\right)}}} \]

      5. metadata-eval 75.4%

         \[\leadsto \sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{\color{blue}{4}}}} \]

   8. Simplified 75.4%

      \[\leadsto \color{blue}{\sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{4}}}} \]

   9. Taylor expanded in k around 0 75.9%

      \[\leadsto \sqrt{\frac{4}{{\left(\frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{4}}} \]

   if 7.9999999999999999e156 < t

   1. Initial program 3.2%

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

   3. Taylor expanded in t around 0 80.9%

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

      1. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

   6. Taylor expanded in k around inf 80.9%

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

      1. times-frac 78.2%

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

   8. Simplified 78.2%

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

      1. associate-*l/ 81.2%

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

      2. associate-/r* 81.2%

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

   10. Applied egg-rr 81.2%

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

   11. Taylor expanded in k around 0 78.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}}{{k}^{2}} \]
   12. Step-by-step derivation

       1. *-commutative 78.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{{k}^{2}} \]

       2. associate-*r/ 78.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)}{{k}^{2}} \]

       3. metadata-eval 78.5%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{\color{blue}{0.16666666666666666}}{t}\right)}{{k}^{2}} \]

   13. Simplified 78.5%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}}{{k}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-80}:\\ \;\;\;\;\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\frac{4}{{\left(k \cdot \frac{\frac{k}{t} \cdot {t}^{1.5}}{\ell}\right)}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} - \frac{0.16666666666666666}{t}\right)}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.6% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l 2.0) (* t_m (pow k_m 2.0))) (pow k_m 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(l, 2.0) / (t_m * pow(k_m, 2.0))) / pow(k_m, 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 * (((l ** 2.0d0) / (t_m * (k_m ** 2.0d0))) / (k_m ** 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(l, 2.0) / (t_m * Math.pow(k_m, 2.0))) / Math.pow(k_m, 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(l, 2.0) / (t_m * math.pow(k_m, 2.0))) / math.pow(k_m, 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(Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 2.0))) / (k_m ^ 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 * (((l ^ 2.0) / (t_m * (k_m ^ 2.0))) / (k_m ^ 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[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Taylor expanded in t around 0 78.3%

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

   1. associate-/l* 79.2%

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

5. Simplified 79.2%

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

6. Taylor expanded in k around inf 78.3%

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

   1. times-frac 78.7%

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

8. Simplified 78.7%

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

   1. associate-*l/ 79.1%

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

   2. associate-/r* 79.2%

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

10. Applied egg-rr 79.2%

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

11. Taylor expanded in k around 0 69.5%

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

12. Final simplification 69.5%

    \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t \cdot {k}^{2}}}{{k}^{2}} \]
13. Add Preprocessing

Alternative 17: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2 \cdot \left({\ell}^{2} \cdot \cos k_m\right)}{t_m \cdot {k_m}^{4}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ (* 2.0 (* (pow l 2.0) (cos k_m))) (* t_m (pow k_m 4.0)))))

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(l, 2.0) * cos(k_m))) / (t_m * pow(k_m, 4.0)));
}

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 * ((l ** 2.0d0) * cos(k_m))) / (t_m * (k_m ** 4.0d0)))
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(l, 2.0) * Math.cos(k_m))) / (t_m * Math.pow(k_m, 4.0)));
}

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(l, 2.0) * math.cos(k_m))) / (t_m * math.pow(k_m, 4.0)))

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(Float64(2.0 * Float64((l ^ 2.0) * cos(k_m))) / Float64(t_m * (k_m ^ 4.0))))
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 * ((l ^ 2.0) * cos(k_m))) / (t_m * (k_m ^ 4.0)));
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[(N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. associate-*l* 31.5%

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

   2. associate-/r* 31.4%

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

   3. sub-neg 31.4%

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

   4. distribute-rgt-in 27.5%

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

   5. unpow2 27.5%

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

   6. times-frac 21.3%

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

   7. sqr-neg 21.3%

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

   8. times-frac 27.5%

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

   9. unpow2 27.5%

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

   10. distribute-rgt-in 31.4%

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

   11. +-commutative 31.4%

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

   12. associate-+l+ 39.8%

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

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 78.3%

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

   1. associate-*r/ 78.3%

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

   2. associate-*r* 78.3%

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

7. Simplified 78.3%

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

8. Taylor expanded in k around 0 67.0%

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

9. Final simplification 67.0%

   \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{t \cdot {k}^{4}} \]
10. Add Preprocessing

Alternative 18: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t_m \cdot {k_m}^{4}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0))))))

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(l, 2.0) / (t_m * pow(k_m, 4.0))));
}

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 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0))))
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(l, 2.0) / (t_m * Math.pow(k_m, 4.0))));
}

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(l, 2.0) / (t_m * math.pow(k_m, 4.0))))

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((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0)))))
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 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0))));
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[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. associate-*l* 31.5%

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

   2. associate-/r* 31.4%

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

   3. sub-neg 31.4%

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

   4. distribute-rgt-in 27.5%

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

   5. unpow2 27.5%

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

   6. times-frac 21.3%

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

   7. sqr-neg 21.3%

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

   8. times-frac 27.5%

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

   9. unpow2 27.5%

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

   10. distribute-rgt-in 31.4%

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

   11. +-commutative 31.4%

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

   12. associate-+l+ 39.8%

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

3. Simplified 39.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 65.2%

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

6. Final simplification 65.2%

   \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))