Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 80.5%

Time: 19.6s

Alternatives: 21

Speedup: 1.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sqrt[3]{k\_m \cdot \sqrt{2}}\\ \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(t\_1 \cdot t\_1\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (cbrt (* k_m (sqrt 2.0)))))
   (if (<= k_m 9.2e-14)
     (/ 2.0 (pow (* (* t (pow (cbrt l) -2.0)) (* t_1 t_1)) 3.0))
     (*
      2.0
      (/
       (* (pow l 2.0) (cos k_m))
       (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cbrt((k_m * sqrt(2.0)));
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / pow(((t * pow(cbrt(l), -2.0)) * (t_1 * t_1)), 3.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cbrt((k_m * Math.sqrt(2.0)));
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * (t_1 * t_1)), 3.0);
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(Float64(k_m * sqrt(2.0)))
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / (Float64(Float64(t * (cbrt(l) ^ -2.0)) * Float64(t_1 * t_1)) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in k around 0 56.0%

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

      1. add-cube-cbrt 55.9%

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

      2. pow3 55.9%

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

      3. cbrt-prod 56.0%

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

      4. associate-/l/ 51.1%

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

      5. unpow2 51.1%

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

      6. cbrt-div 51.6%

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

      7. unpow3 51.6%

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

      8. add-cbrt-cube 60.1%

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

      9. unpow2 60.1%

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

      10. cbrt-prod 64.8%

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

      11. unpow2 64.8%

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

      12. div-inv 64.8%

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

      13. pow-flip 64.8%

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

      14. metadata-eval 64.8%

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

   7. Applied egg-rr 64.8%

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

      1. pow1/3 64.2%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \color{blue}{{\left(2 \cdot {k}^{2}\right)}^{0.3333333333333333}}\right)}^{3}} \]

      2. add-sqr-sqrt 64.2%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot {\color{blue}{\left(\sqrt{2 \cdot {k}^{2}} \cdot \sqrt{2 \cdot {k}^{2}}\right)}}^{0.3333333333333333}\right)}^{3}} \]

      3. unpow-prod-down 64.2%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \color{blue}{\left({\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

      4. *-commutative 64.2%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt{\color{blue}{{k}^{2} \cdot 2}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      5. sqrt-prod 64.2%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\color{blue}{\left(\sqrt{{k}^{2}} \cdot \sqrt{2}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      6. sqrt-pow1 18.1%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\color{blue}{{k}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      7. metadata-eval 18.1%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left({k}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      8. pow1 18.1%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\color{blue}{k} \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      9. *-commutative 18.1%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\color{blue}{{k}^{2} \cdot 2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      10. sqrt-prod 18.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\color{blue}{\left(\sqrt{{k}^{2}} \cdot \sqrt{2}\right)}}^{0.3333333333333333}\right)\right)}^{3}} \]

      11. sqrt-pow1 24.3%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\color{blue}{{k}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{2}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      12. metadata-eval 24.3%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left({k}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      13. pow1 24.3%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\color{blue}{k} \cdot \sqrt{2}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

   9. Applied egg-rr 24.3%

      \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \color{blue}{\left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]
   10. Step-by-step derivation

       1. unpow1/3 24.4%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\color{blue}{\sqrt[3]{k \cdot \sqrt{2}}} \cdot {\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

       2. unpow1/3 77.3%

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

   11. Simplified 77.3%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 78.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{k \cdot \sqrt{2}} \cdot \sqrt[3]{k \cdot \sqrt{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \frac{\sin k\_m}{\cos k\_m}\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k\_m}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.1e+16)
   (/
    2.0
    (*
     (* 2.0 (/ (sin k_m) (cos k_m)))
     (pow (* (* t (pow (cbrt l) -2.0)) (cbrt (sin k_m))) 3.0)))
   (*
    2.0
    (/
     (* (pow l 2.0) (cos k_m))
     (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.1e+16) {
		tmp = 2.0 / ((2.0 * (sin(k_m) / cos(k_m))) * pow(((t * pow(cbrt(l), -2.0)) * cbrt(sin(k_m))), 3.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.1e+16) {
		tmp = 2.0 / ((2.0 * (Math.sin(k_m) / Math.cos(k_m))) * Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k_m))), 3.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.1e+16)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(sin(k_m) / cos(k_m))) * (Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(sin(k_m))) ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.1e+16], N[(2.0 / N[(N[(2.0 * N[(N[Sin[k$95$m], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.1e16

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

      1. add-cube-cbrt 55.6%

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

      2. pow3 55.6%

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

      3. *-commutative 55.6%

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

      4. cbrt-prod 55.6%

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

      5. cbrt-div 56.5%

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

      6. rem-cbrt-cube 67.5%

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

      7. cbrt-prod 79.5%

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

      8. pow2 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. *-commutative 79.5%

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

   8. Simplified 79.5%

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

      1. pow1 79.5%

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

      2. div-inv 79.5%

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

      3. pow-flip 79.4%

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

      4. metadata-eval 79.4%

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

   10. Applied egg-rr 79.4%

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

       1. unpow1 79.4%

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

   12. Simplified 79.4%

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

   13. Taylor expanded in t around inf 73.0%

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

   if 3.1e16 < k

   1. Initial program 49.3%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around 0 76.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \frac{\sin k}{\cos k}\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(2 \cdot \frac{\sin k\_m}{\cos k\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e+16)
   (/
    2.0
    (*
     (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0)
     (* 2.0 (/ (sin k_m) (cos k_m)))))
   (*
    2.0
    (/
     (* (pow l 2.0) (cos k_m))
     (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e+16) {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (2.0 * (sin(k_m) / cos(k_m))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e+16) {
		tmp = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (2.0 * (Math.sin(k_m) / Math.cos(k_m))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e+16)
		tmp = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(2.0 * Float64(sin(k_m) / cos(k_m)))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.2e+16], N[(2.0 / N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[(N[Sin[k$95$m], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e16

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

      1. add-cube-cbrt 55.6%

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

      2. pow3 55.6%

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

      3. *-commutative 55.6%

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

      4. cbrt-prod 55.6%

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

      5. cbrt-div 56.5%

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

      6. rem-cbrt-cube 67.5%

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

      7. cbrt-prod 79.5%

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

      8. pow2 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. *-commutative 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in t around inf 73.1%

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

   if 2.2e16 < k

   1. Initial program 49.3%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around 0 76.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-14)
   (/
    2.0
    (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0) (* k_m 2.0)))
   (*
    2.0
    (/
     (* (pow l 2.0) (cos k_m))
     (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k_m)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k_m)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k_m)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 55.9%

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

      3. *-commutative 55.9%

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

      4. cbrt-prod 55.9%

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

      5. cbrt-div 56.9%

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

      6. rem-cbrt-cube 68.0%

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

      7. cbrt-prod 80.2%

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

      8. pow2 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

   8. Simplified 80.2%

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

   9. Taylor expanded in k around 0 71.8%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 78.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-14)
   (/
    2.0
    (* (pow (* (* t (pow (cbrt l) -2.0)) (cbrt (sin k_m))) 3.0) (* k_m 2.0)))
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / (pow(((t * pow(cbrt(l), -2.0)) * cbrt(sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / (Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / Float64((Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[(N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 55.9%

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

      3. *-commutative 55.9%

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

      4. cbrt-prod 55.9%

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

      5. cbrt-div 56.9%

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

      6. rem-cbrt-cube 68.0%

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

      7. cbrt-prod 80.2%

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

      8. pow2 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

   8. Simplified 80.2%

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

      1. pow1 80.2%

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

      2. div-inv 80.2%

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

      3. pow-flip 80.1%

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

      4. metadata-eval 80.1%

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

   10. Applied egg-rr 80.1%

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

       1. unpow1 80.1%

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

   12. Simplified 80.1%

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

   13. Taylor expanded in k around 0 71.7%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 77.9%

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

      1. unpow2 77.9%

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

      2. sin-mult 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. div-sub 77.9%

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

      2. +-inverses 77.9%

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

      3. cos-0 77.9%

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

      4. metadata-eval 77.9%

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

      5. count-2 77.9%

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

      6. *-commutative 77.9%

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

   9. Simplified 77.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-14)
   (/
    2.0
    (* (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0) (* k_m 2.0)))
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / (pow(((t / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / (Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / Float64((Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(k_m * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[(N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 55.9%

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

      3. *-commutative 55.9%

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

      4. cbrt-prod 55.9%

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

      5. cbrt-div 56.9%

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

      6. rem-cbrt-cube 68.0%

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

      7. cbrt-prod 80.2%

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

      8. pow2 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

   8. Simplified 80.2%

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

   9. Taylor expanded in k around 0 71.8%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 77.9%

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

      1. unpow2 77.9%

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

      2. sin-mult 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. div-sub 77.9%

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

      2. +-inverses 77.9%

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

      3. cos-0 77.9%

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

      4. metadata-eval 77.9%

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

      5. count-2 77.9%

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

      6. *-commutative 77.9%

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

   9. Simplified 77.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\\ \mathbf{if}\;t \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k\_m}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + \frac{k\_m}{t \cdot \frac{t}{k\_m}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (/ (pow t 3.0) (* l l)))))
   (if (<= t 1.25e-75)
     (pow (* l (sqrt (/ (/ 2.0 t) (pow k_m 4.0)))) 2.0)
     (if (<= t 2.8e+102)
       (/ 2.0 (* t_1 (* (tan k_m) (+ 1.0 (+ 1.0 (/ k_m (* t (/ t k_m))))))))
       (if (<= t 8.5e+165)
         (/ 2.0 (* (pow (* t (pow (cbrt l) -2.0)) 3.0) (* 2.0 (pow k_m 2.0))))
         (/ 2.0 (* (* k_m 2.0) t_1)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * (pow(t, 3.0) / (l * l));
	double tmp;
	if (t <= 1.25e-75) {
		tmp = pow((l * sqrt(((2.0 / t) / pow(k_m, 4.0)))), 2.0);
	} else if (t <= 2.8e+102) {
		tmp = 2.0 / (t_1 * (tan(k_m) * (1.0 + (1.0 + (k_m / (t * (t / k_m)))))));
	} else if (t <= 8.5e+165) {
		tmp = 2.0 / (pow((t * pow(cbrt(l), -2.0)), 3.0) * (2.0 * pow(k_m, 2.0)));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * t_1);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l));
	double tmp;
	if (t <= 1.25e-75) {
		tmp = Math.pow((l * Math.sqrt(((2.0 / t) / Math.pow(k_m, 4.0)))), 2.0);
	} else if (t <= 2.8e+102) {
		tmp = 2.0 / (t_1 * (Math.tan(k_m) * (1.0 + (1.0 + (k_m / (t * (t / k_m)))))));
	} else if (t <= 8.5e+165) {
		tmp = 2.0 / (Math.pow((t * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * Math.pow(k_m, 2.0)));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))
	tmp = 0.0
	if (t <= 1.25e-75)
		tmp = Float64(l * sqrt(Float64(Float64(2.0 / t) / (k_m ^ 4.0)))) ^ 2.0;
	elseif (t <= 2.8e+102)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + Float64(k_m / Float64(t * Float64(t / k_m))))))));
	elseif (t <= 8.5e+165)
		tmp = Float64(2.0 / Float64((Float64(t * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * (k_m ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.25e-75], N[Power[N[(l * N[Sqrt[N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 2.8e+102], N[(2.0 / N[(t$95$1 * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k$95$m / N[(t * N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e+165], N[(2.0 / N[(N[Power[N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.24999999999999995e-75

   1. Initial program 47.1%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 42.9%

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

      1. associate-*r/ 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. times-frac 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in t around 0 51.7%

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

      1. associate-*r/ 51.7%

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

      2. *-commutative 51.7%

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

      3. associate-/l* 51.7%

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

      4. *-commutative 51.7%

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

   10. Simplified 51.7%

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

       1. add-sqr-sqrt 34.8%

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

       2. pow2 34.8%

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

       3. sqrt-prod 31.6%

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

       4. sqrt-pow1 35.1%

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

       5. metadata-eval 35.1%

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

       6. pow1 35.1%

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

       7. associate-/r* 35.1%

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

   12. Applied egg-rr 35.1%

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

   if 1.24999999999999995e-75 < t < 2.80000000000000018e102

   1. Initial program 74.0%

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

   3. Simplified 73.9%

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

      1. unpow2 73.9%

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

      2. clear-num 73.9%

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

      3. frac-times 74.0%

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

      4. *-un-lft-identity 74.0%

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

   6. Applied egg-rr 74.0%

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

   if 2.80000000000000018e102 < t < 8.5000000000000001e165

   1. Initial program 56.1%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in k around 0 56.0%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 56.0%

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

      3. associate-/l/ 55.9%

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

      4. unpow2 55.9%

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

      5. cbrt-div 55.9%

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

      6. unpow3 55.9%

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

      7. add-cbrt-cube 82.2%

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

      8. unpow2 82.2%

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

      9. cbrt-prod 82.0%

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

      10. unpow2 82.0%

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

      11. div-inv 81.7%

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

      12. unpow-prod-down 55.9%

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

      13. pow-flip 55.9%

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

      14. metadata-eval 55.9%

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

   7. Applied egg-rr 55.9%

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

      1. cube-prod 81.7%

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

   9. Simplified 81.7%

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

   if 8.5000000000000001e165 < t

   1. Initial program 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in k around 0 80.0%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k\_m \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-14)
   (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 56.0%

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

      3. cbrt-div 55.9%

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

      4. rem-cbrt-cube 65.2%

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

      5. cbrt-prod 74.6%

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

      6. pow2 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in k around 0 66.1%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 77.9%

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

      1. unpow2 77.9%

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

      2. sin-mult 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. div-sub 77.9%

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

      2. +-inverses 77.9%

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

      3. cos-0 77.9%

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

      4. metadata-eval 77.9%

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

      5. count-2 77.9%

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

      6. *-commutative 77.9%

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

   9. Simplified 77.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t \cdot {k\_m}^{2}\right)}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-14)
   (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
   (/
    2.0
    (/ (* (pow k_m 2.0) (* t (pow k_m 2.0))) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * pow(k_m, 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * Math.pow(k_m, 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * (k_m ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 56.0%

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

      3. cbrt-div 55.9%

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

      4. rem-cbrt-cube 65.2%

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

      5. cbrt-prod 74.6%

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

      6. pow2 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in k around 0 66.1%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 77.9%

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

   6. Taylor expanded in k around 0 63.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2} \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-14)
   (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-14) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-14)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-14], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999993e-14

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. add-cube-cbrt 56.0%

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

      2. pow3 56.0%

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

      3. cbrt-div 55.9%

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

      4. rem-cbrt-cube 65.2%

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

      5. cbrt-prod 74.6%

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

      6. pow2 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in k around 0 66.1%

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

   if 9.19999999999999993e-14 < k

   1. Initial program 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in t around 0 77.9%

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

   6. Taylor expanded in k around 0 61.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\\ \mathbf{if}\;t \leq 1.45 \cdot 10^{-75}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k\_m}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + \frac{k\_m}{t \cdot \frac{t}{k\_m}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (/ (pow t 3.0) (* l l)))))
   (if (<= t 1.45e-75)
     (pow (* l (sqrt (/ (/ 2.0 t) (pow k_m 4.0)))) 2.0)
     (if (<= t 5.3e+101)
       (/ 2.0 (* t_1 (* (tan k_m) (+ 1.0 (+ 1.0 (/ k_m (* t (/ t k_m))))))))
       (if (<= t 3.2e+159)
         (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (pow (/ t (cbrt l)) 3.0) l)))
         (/ 2.0 (* (* k_m 2.0) t_1)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * (pow(t, 3.0) / (l * l));
	double tmp;
	if (t <= 1.45e-75) {
		tmp = pow((l * sqrt(((2.0 / t) / pow(k_m, 4.0)))), 2.0);
	} else if (t <= 5.3e+101) {
		tmp = 2.0 / (t_1 * (tan(k_m) * (1.0 + (1.0 + (k_m / (t * (t / k_m)))))));
	} else if (t <= 3.2e+159) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * (pow((t / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * t_1);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l));
	double tmp;
	if (t <= 1.45e-75) {
		tmp = Math.pow((l * Math.sqrt(((2.0 / t) / Math.pow(k_m, 4.0)))), 2.0);
	} else if (t <= 5.3e+101) {
		tmp = 2.0 / (t_1 * (Math.tan(k_m) * (1.0 + (1.0 + (k_m / (t * (t / k_m)))))));
	} else if (t <= 3.2e+159) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * (Math.pow((t / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))
	tmp = 0.0
	if (t <= 1.45e-75)
		tmp = Float64(l * sqrt(Float64(Float64(2.0 / t) / (k_m ^ 4.0)))) ^ 2.0;
	elseif (t <= 5.3e+101)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + Float64(k_m / Float64(t * Float64(t / k_m))))))));
	elseif (t <= 3.2e+159)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.45e-75], N[Power[N[(l * N[Sqrt[N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 5.3e+101], N[(2.0 / N[(t$95$1 * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(k$95$m / N[(t * N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+159], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.4500000000000001e-75

   1. Initial program 47.1%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 42.9%

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

      1. associate-*r/ 42.9%

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

      2. *-commutative 42.9%

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

      3. *-commutative 42.9%

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

      4. times-frac 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in t around 0 51.7%

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

      1. associate-*r/ 51.7%

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

      2. *-commutative 51.7%

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

      3. associate-/l* 51.7%

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

      4. *-commutative 51.7%

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

   10. Simplified 51.7%

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

       1. add-sqr-sqrt 34.8%

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

       2. pow2 34.8%

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

       3. sqrt-prod 31.6%

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

       4. sqrt-pow1 35.1%

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

       5. metadata-eval 35.1%

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

       6. pow1 35.1%

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

       7. associate-/r* 35.1%

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

   12. Applied egg-rr 35.1%

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

   if 1.4500000000000001e-75 < t < 5.30000000000000006e101

   1. Initial program 74.0%

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

   3. Simplified 73.9%

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

      1. unpow2 73.9%

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

      2. clear-num 73.9%

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

      3. frac-times 74.0%

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

      4. *-un-lft-identity 74.0%

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

   6. Applied egg-rr 74.0%

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

   if 5.30000000000000006e101 < t < 3.19999999999999985e159

   1. Initial program 51.7%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in k around 0 51.6%

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

      1. add-cube-cbrt 51.6%

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

      2. pow3 51.6%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 80.0%

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

   7. Applied egg-rr 80.0%

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

   if 3.19999999999999985e159 < t

   1. Initial program 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 80.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-75}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.65 \cdot 10^{-96}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k\_m}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + \frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}\right)\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.65e-96)
   (pow (* l (sqrt (/ (/ 2.0 t) (pow k_m 4.0)))) 2.0)
   (if (<= t 5.3e+101)
     (/
      2.0
      (*
       (/ (/ (pow t 3.0) l) l)
       (* (* (sin k_m) (tan k_m)) (+ 2.0 (/ (/ k_m t) (/ t k_m))))))
     (if (<= t 3.5e+159)
       (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (pow (/ t (cbrt l)) 3.0) l)))
       (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t 3.0) (* l l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.65e-96) {
		tmp = pow((l * sqrt(((2.0 / t) / pow(k_m, 4.0)))), 2.0);
	} else if (t <= 5.3e+101) {
		tmp = 2.0 / (((pow(t, 3.0) / l) / l) * ((sin(k_m) * tan(k_m)) * (2.0 + ((k_m / t) / (t / k_m)))));
	} else if (t <= 3.5e+159) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * (pow((t / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.65e-96) {
		tmp = Math.pow((l * Math.sqrt(((2.0 / t) / Math.pow(k_m, 4.0)))), 2.0);
	} else if (t <= 5.3e+101) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) / l) * ((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + ((k_m / t) / (t / k_m)))));
	} else if (t <= 3.5e+159) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * (Math.pow((t / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.65e-96)
		tmp = Float64(l * sqrt(Float64(Float64(2.0 / t) / (k_m ^ 4.0)))) ^ 2.0;
	elseif (t <= 5.3e+101)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) / l) * Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + Float64(Float64(k_m / t) / Float64(t / k_m))))));
	elseif (t <= 3.5e+159)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.65e-96], N[Power[N[(l * N[Sqrt[N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 5.3e+101], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e+159], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.64999999999999997e-96

   1. Initial program 46.7%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in k around 0 42.8%

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

      1. associate-*r/ 42.8%

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

      2. *-commutative 42.8%

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

      3. *-commutative 42.8%

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

      4. times-frac 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in t around 0 52.0%

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

      1. associate-*r/ 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-/l* 52.0%

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

      4. *-commutative 52.0%

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

   10. Simplified 52.0%

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

       1. add-sqr-sqrt 34.4%

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

       2. pow2 34.4%

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

       3. sqrt-prod 31.1%

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

       4. sqrt-pow1 34.2%

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

       5. metadata-eval 34.2%

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

       6. pow1 34.2%

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

       7. associate-/r* 34.1%

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

   12. Applied egg-rr 34.1%

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

   if 3.64999999999999997e-96 < t < 5.30000000000000006e101

   1. Initial program 70.8%

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

   3. Simplified 66.0%

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

      1. unpow2 66.0%

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

      2. clear-num 66.0%

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

      3. un-div-inv 66.0%

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

   6. Applied egg-rr 66.0%

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

   if 5.30000000000000006e101 < t < 3.4999999999999999e159

   1. Initial program 51.7%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in k around 0 51.6%

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

      1. add-cube-cbrt 51.6%

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

      2. pow3 51.6%

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

      3. cbrt-div 51.6%

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

      4. rem-cbrt-cube 80.0%

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

   7. Applied egg-rr 80.0%

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

   if 3.4999999999999999e159 < t

   1. Initial program 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 80.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.65 \cdot 10^{-96}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k\_m}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.05e-19)
   (pow (* l (sqrt (/ (/ 2.0 t) (pow k_m 4.0)))) 2.0)
   (if (<= t 3.9e+158)
     (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (pow (/ t (cbrt l)) 3.0) l)))
     (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t 3.0) (* l l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-19) {
		tmp = pow((l * sqrt(((2.0 / t) / pow(k_m, 4.0)))), 2.0);
	} else if (t <= 3.9e+158) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * (pow((t / cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-19) {
		tmp = Math.pow((l * Math.sqrt(((2.0 / t) / Math.pow(k_m, 4.0)))), 2.0);
	} else if (t <= 3.9e+158) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * (Math.pow((t / Math.cbrt(l)), 3.0) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.05e-19)
		tmp = Float64(l * sqrt(Float64(Float64(2.0 / t) / (k_m ^ 4.0)))) ^ 2.0;
	elseif (t <= 3.9e+158)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((Float64(t / cbrt(l)) ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.05e-19], N[Power[N[(l * N[Sqrt[N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 3.9e+158], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.0499999999999999e-19

   1. Initial program 48.0%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in k around 0 43.1%

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

      1. associate-*r/ 43.1%

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

      2. *-commutative 43.1%

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

      3. *-commutative 43.1%

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

      4. times-frac 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in t around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-/l* 51.5%

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

      4. *-commutative 51.5%

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

   10. Simplified 51.5%

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

       1. add-sqr-sqrt 35.5%

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

       2. pow2 35.5%

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

       3. sqrt-prod 32.4%

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

       4. sqrt-pow1 35.8%

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

       5. metadata-eval 35.8%

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

       6. pow1 35.8%

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

       7. associate-/r* 35.8%

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

   12. Applied egg-rr 35.8%

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

   if 1.0499999999999999e-19 < t < 3.9e158

   1. Initial program 69.9%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. add-cube-cbrt 53.7%

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

      2. pow3 53.7%

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

      3. cbrt-div 53.7%

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

      4. rem-cbrt-cube 63.5%

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

   7. Applied egg-rr 63.5%

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

   if 3.9e158 < t

   1. Initial program 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 80.7%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+158}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k\_m}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{2} \cdot \left(t \cdot \frac{1}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.05e-19)
   (pow (* l (sqrt (/ (/ 2.0 t) (pow k_m 4.0)))) 2.0)
   (if (<= t 3.6e+158)
     (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (* (pow t 2.0) (* t (/ 1.0 l))) l)))
     (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t 3.0) (* l l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-19) {
		tmp = pow((l * sqrt(((2.0 / t) / pow(k_m, 4.0)))), 2.0);
	} else if (t <= 3.6e+158) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t, 2.0) * (t * (1.0 / l))) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.05d-19) then
        tmp = (l * sqrt(((2.0d0 / t) / (k_m ** 4.0d0)))) ** 2.0d0
    else if (t <= 3.6d+158) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t ** 2.0d0) * (t * (1.0d0 / l))) / l))
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t ** 3.0d0) / (l * l))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-19) {
		tmp = Math.pow((l * Math.sqrt(((2.0 / t) / Math.pow(k_m, 4.0)))), 2.0);
	} else if (t <= 3.6e+158) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t, 2.0) * (t * (1.0 / l))) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t, 3.0) / (l * l))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.05e-19:
		tmp = math.pow((l * math.sqrt(((2.0 / t) / math.pow(k_m, 4.0)))), 2.0)
	elif t <= 3.6e+158:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t, 2.0) * (t * (1.0 / l))) / l))
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t, 3.0) / (l * l))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.05e-19)
		tmp = Float64(l * sqrt(Float64(Float64(2.0 / t) / (k_m ^ 4.0)))) ^ 2.0;
	elseif (t <= 3.6e+158)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t ^ 2.0) * Float64(t * Float64(1.0 / l))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t ^ 3.0) / Float64(l * l)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.05e-19)
		tmp = (l * sqrt(((2.0 / t) / (k_m ^ 4.0)))) ^ 2.0;
	elseif (t <= 3.6e+158)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t ^ 2.0) * (t * (1.0 / l))) / l));
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t ^ 3.0) / (l * l))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.05e-19], N[Power[N[(l * N[Sqrt[N[(N[(2.0 / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 3.6e+158], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] * N[(t * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.0499999999999999e-19

   1. Initial program 48.0%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in k around 0 43.1%

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

      1. associate-*r/ 43.1%

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

      2. *-commutative 43.1%

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

      3. *-commutative 43.1%

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

      4. times-frac 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in t around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-/l* 51.5%

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

      4. *-commutative 51.5%

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

   10. Simplified 51.5%

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

       1. add-sqr-sqrt 35.5%

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

       2. pow2 35.5%

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

       3. sqrt-prod 32.4%

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

       4. sqrt-pow1 35.8%

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

       5. metadata-eval 35.8%

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

       6. pow1 35.8%

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

       7. associate-/r* 35.8%

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

   12. Applied egg-rr 35.8%

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

   if 1.0499999999999999e-19 < t < 3.59999999999999988e158

   1. Initial program 69.9%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. div-inv 53.8%

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

      2. unpow3 53.8%

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

      3. associate-*l* 60.4%

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

      4. pow2 60.4%

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

   7. Applied egg-rr 60.4%

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

   if 3.59999999999999988e158 < t

   1. Initial program 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in k around 0 80.7%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{\frac{2}{t}}{{k}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+158}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \left(t \cdot \frac{1}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{2} \cdot \left(t \cdot \frac{1}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e+23)
   (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (* (pow t 2.0) (* t (/ 1.0 l))) l)))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e+23) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t, 2.0) * (t * (1.0 / l))) / l));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.2d+23) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t ** 2.0d0) * (t * (1.0d0 / l))) / l))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e+23) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t, 2.0) * (t * (1.0 / l))) / l));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.2e+23:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t, 2.0) * (t * (1.0 / l))) / l))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e+23)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t ^ 2.0) * Float64(t * Float64(1.0 / l))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.2e+23)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t ^ 2.0) * (t * (1.0 / l))) / l));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.2e+23], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] * N[(t * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000008e23

   1. Initial program 55.7%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in k around 0 56.1%

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

      1. div-inv 56.1%

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

      2. unpow3 56.1%

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

      3. associate-*l* 59.5%

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

      4. pow2 59.5%

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

   7. Applied egg-rr 59.5%

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

   if 2.20000000000000008e23 < k

   1. Initial program 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around 0 75.9%

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

   6. Taylor expanded in k around 0 59.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{2} \cdot \left(t \cdot \frac{1}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.04 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.04e+24)
   (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (/ (pow t 3.0) l) l)))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.04e+24) {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t, 3.0) / l) / l));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.04d+24) then
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t ** 3.0d0) / l) / l))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.04e+24) {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t, 3.0) / l) / l));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.04e+24:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t, 3.0) / l) / l))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.04e+24)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t ^ 3.0) / l) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.04e+24)
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t ^ 3.0) / l) / l));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.04e+24], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.03999999999999997e24

   1. Initial program 55.7%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in k around 0 56.1%

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

   if 1.03999999999999997e24 < k

   1. Initial program 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around 0 75.9%

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

   6. Taylor expanded in k around 0 59.5%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.04 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k\_m}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e+23)
   (/ 2.0 (* (/ (pow t 3.0) l) (/ (* 2.0 (pow k_m 2.0)) l)))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e+23) {
		tmp = 2.0 / ((pow(t, 3.0) / l) * ((2.0 * pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.5d+23) then
        tmp = 2.0d0 / (((t ** 3.0d0) / l) * ((2.0d0 * (k_m ** 2.0d0)) / l))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e+23) {
		tmp = 2.0 / ((Math.pow(t, 3.0) / l) * ((2.0 * Math.pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.5e+23:
		tmp = 2.0 / ((math.pow(t, 3.0) / l) * ((2.0 * math.pow(k_m, 2.0)) / l))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e+23)
		tmp = Float64(2.0 / Float64(Float64((t ^ 3.0) / l) * Float64(Float64(2.0 * (k_m ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.5e+23)
		tmp = 2.0 / (((t ^ 3.0) / l) * ((2.0 * (k_m ^ 2.0)) / l));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.5e+23], N[(2.0 / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999987e23

   1. Initial program 55.7%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in k around 0 56.1%

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

      1. associate-*l/ 57.1%

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

   7. Applied egg-rr 57.1%

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

      1. associate-/l* 56.6%

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

   9. Simplified 56.6%

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

   if 7.49999999999999987e23 < k

   1. Initial program 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around 0 75.9%

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

   6. Taylor expanded in k around 0 59.5%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k\_m}^{4}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.45e+23)
   (/ 2.0 (/ (* (* 2.0 (pow k_m 2.0)) (/ (pow t 3.0) l)) l))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (pow l 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e+23) {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / pow(l, 2.0));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.45d+23) then
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / (l ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.45e+23) {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / Math.pow(l, 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.45e+23:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / math.pow(l, 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.45e+23)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / (l ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.45e+23)
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / (l ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.45e+23], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000001e23

   1. Initial program 55.7%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in k around 0 56.1%

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

      1. associate-*l/ 57.1%

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

   7. Applied egg-rr 57.1%

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

   if 2.4500000000000001e23 < k

   1. Initial program 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in t around 0 75.9%

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

   6. Taylor expanded in k around 0 59.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{2}{\frac{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{4}} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * (pow(l, 2.0) / (t * pow(k_m, 4.0)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 * ((l ** 2.0d0) / (t * (k_m ** 4.0d0)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 4.0)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * (math.pow(l, 2.0) / (t * math.pow(k_m, 4.0)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64((l ^ 2.0) / Float64(t * (k_m ^ 4.0))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * ((l ^ 2.0) / (t * (k_m ^ 4.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

3. Simplified 50.7%

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

5. Taylor expanded in k around 0 47.9%

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

   1. associate-*r/ 47.9%

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

   2. *-commutative 47.9%

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

   3. *-commutative 47.9%

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

   4. times-frac 47.4%

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

7. Simplified 47.4%

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

8. Taylor expanded in t around 0 52.4%

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

9. Final simplification 52.4%

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

Alternative 20: 52.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{{k\_m}^{4} \cdot \frac{t}{{\ell}^{2}}} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (pow(k_m, 4.0) * (t / pow(l, 2.0)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m ** 4.0d0) * (t / (l ** 2.0d0)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / (Math.pow(k_m, 4.0) * (t / Math.pow(l, 2.0)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (math.pow(k_m, 4.0) * (t / math.pow(l, 2.0)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t / (l ^ 2.0))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((k_m ^ 4.0) * (t / (l ^ 2.0)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

3. Simplified 54.3%

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

5. Taylor expanded in t around 0 61.4%

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

6. Taylor expanded in k around 0 52.4%

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

   1. associate-/l* 52.7%

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

8. Simplified 52.7%

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

9. Final simplification 52.7%

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

Alternative 21: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{2 \cdot {\ell}^{2}}{t}}{{k\_m}^{4}} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((2.0 * pow(l, 2.0)) / t) / pow(k_m, 4.0);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((2.0d0 * (l ** 2.0d0)) / t) / (k_m ** 4.0d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((2.0 * Math.pow(l, 2.0)) / t) / Math.pow(k_m, 4.0);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((2.0 * math.pow(l, 2.0)) / t) / math.pow(k_m, 4.0)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(2.0 * (l ^ 2.0)) / t) / (k_m ^ 4.0))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 * (l ^ 2.0)) / t) / (k_m ^ 4.0);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

3. Simplified 50.7%

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

5. Taylor expanded in k around 0 47.9%

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

   1. associate-*r/ 47.9%

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

   2. *-commutative 47.9%

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

   3. *-commutative 47.9%

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

   4. times-frac 47.4%

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

7. Simplified 47.4%

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

8. Taylor expanded in t around 0 52.4%

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

   1. pow2 52.4%

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

   2. associate-*r/ 52.4%

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

   3. pow2 52.4%

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

   4. *-commutative 52.4%

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

10. Applied egg-rr 52.4%

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

    1. associate-/r* 52.7%

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

12. Simplified 52.7%

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

13. Final simplification 52.7%

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

Reproduce

herbie shell --seed 2024082 
(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))))