Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 91.4%

Time: 19.3s

Alternatives: 18

Speedup: 3.8×

• 40.8% 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: 35.5% 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: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}\\ \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{t\_1}{k\_m \cdot \left(\sqrt[3]{k\_m \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k\_m}^{7}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_1}{t \cdot \left({\left(\sqrt[3]{\frac{k\_m}{t}}\right)}^{2} \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (pow (cbrt l) 2.0) (cbrt 2.0))))
   (if (<= k_m 7.2e-6)
     (pow
      (/
       t_1
       (*
        k_m
        (+
         (cbrt (* k_m t))
         (* 0.05555555555555555 (cbrt (* t (pow k_m 7.0)))))))
      3.0)
     (pow
      (/
       t_1
       (* t (* (pow (cbrt (/ k_m t)) 2.0) (cbrt (* (sin k_m) (tan k_m))))))
      3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), 2.0) * cbrt(2.0);
	double tmp;
	if (k_m <= 7.2e-6) {
		tmp = pow((t_1 / (k_m * (cbrt((k_m * t)) + (0.05555555555555555 * cbrt((t * pow(k_m, 7.0))))))), 3.0);
	} else {
		tmp = pow((t_1 / (t * (pow(cbrt((k_m / t)), 2.0) * cbrt((sin(k_m) * tan(k_m)))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(2.0);
	double tmp;
	if (k_m <= 7.2e-6) {
		tmp = Math.pow((t_1 / (k_m * (Math.cbrt((k_m * t)) + (0.05555555555555555 * Math.cbrt((t * Math.pow(k_m, 7.0))))))), 3.0);
	} else {
		tmp = Math.pow((t_1 / (t * (Math.pow(Math.cbrt((k_m / t)), 2.0) * Math.cbrt((Math.sin(k_m) * Math.tan(k_m)))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64((cbrt(l) ^ 2.0) * cbrt(2.0))
	tmp = 0.0
	if (k_m <= 7.2e-6)
		tmp = Float64(t_1 / Float64(k_m * Float64(cbrt(Float64(k_m * t)) + Float64(0.05555555555555555 * cbrt(Float64(t * (k_m ^ 7.0))))))) ^ 3.0;
	else
		tmp = Float64(t_1 / Float64(t * Float64((cbrt(Float64(k_m / t)) ^ 2.0) * cbrt(Float64(sin(k_m) * tan(k_m)))))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 7.2e-6], N[Power[N[(t$95$1 / N[(k$95$m * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 1/3], $MachinePrecision] + N[(0.05555555555555555 * N[Power[N[(t * N[Power[k$95$m, 7.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(t$95$1 / N[(t * N[(N[Power[N[Power[N[(k$95$m / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999967e-6

   1. Initial program 38.4%

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

   3. Simplified 43.7%

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

      1. add-cube-cbrt 43.7%

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

      2. pow3 43.7%

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

   6. Applied egg-rr 57.8%

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

      1. pow2 57.8%

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

      2. *-commutative 57.8%

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

      3. cbrt-prod 57.7%

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

      4. cbrt-prod 70.4%

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

      5. unpow2 70.4%

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

   8. Applied egg-rr 70.4%

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

   9. Taylor expanded in k around 0 83.5%

      \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\color{blue}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{{k}^{7} \cdot t}\right)}}\right)}^{3} \]

   if 7.19999999999999967e-6 < k

   1. Initial program 27.0%

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

   3. Simplified 34.5%

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

      1. add-cube-cbrt 34.4%

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

      2. pow3 34.4%

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

   6. Applied egg-rr 56.2%

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

      1. pow2 56.2%

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

      2. *-commutative 56.2%

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

      3. cbrt-prod 56.1%

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

      4. cbrt-prod 67.5%

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

      5. unpow2 67.5%

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

   8. Applied egg-rr 67.5%

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

      1. cbrt-prod 67.5%

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

      2. unpow2 67.5%

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

      3. cbrt-prod 82.9%

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

      4. pow2 82.9%

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

   10. Applied egg-rr 82.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k}^{7}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{t \cdot \left({\left(\sqrt[3]{\frac{k}{t}}\right)}^{2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}\\ \mathbf{if}\;k\_m \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{t\_1}{k\_m \cdot \left(\sqrt[3]{k\_m \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k\_m}^{7}}\right)}\right)}^{3}\\ \mathbf{elif}\;k\_m \leq 1.46 \cdot 10^{+158}:\\ \;\;\;\;{\left(\frac{t\_1}{\sqrt[3]{\frac{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}{\cos k\_m}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_1}{t \cdot \sqrt[3]{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{k\_m}{t} \cdot \frac{k\_m}{t}\right)}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (pow (cbrt l) 2.0) (cbrt 2.0))))
   (if (<= k_m 4e-9)
     (pow
      (/
       t_1
       (*
        k_m
        (+
         (cbrt (* k_m t))
         (* 0.05555555555555555 (cbrt (* t (pow k_m 7.0)))))))
      3.0)
     (if (<= k_m 1.46e+158)
       (pow
        (/
         t_1
         (cbrt (/ (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))) (cos k_m))))
        3.0)
       (pow
        (/
         t_1
         (* t (cbrt (* (* (sin k_m) (tan k_m)) (* (/ k_m t) (/ k_m t))))))
        3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), 2.0) * cbrt(2.0);
	double tmp;
	if (k_m <= 4e-9) {
		tmp = pow((t_1 / (k_m * (cbrt((k_m * t)) + (0.05555555555555555 * cbrt((t * pow(k_m, 7.0))))))), 3.0);
	} else if (k_m <= 1.46e+158) {
		tmp = pow((t_1 / cbrt(((pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))) / cos(k_m)))), 3.0);
	} else {
		tmp = pow((t_1 / (t * cbrt(((sin(k_m) * tan(k_m)) * ((k_m / t) * (k_m / t)))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(2.0);
	double tmp;
	if (k_m <= 4e-9) {
		tmp = Math.pow((t_1 / (k_m * (Math.cbrt((k_m * t)) + (0.05555555555555555 * Math.cbrt((t * Math.pow(k_m, 7.0))))))), 3.0);
	} else if (k_m <= 1.46e+158) {
		tmp = Math.pow((t_1 / Math.cbrt(((Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))) / Math.cos(k_m)))), 3.0);
	} else {
		tmp = Math.pow((t_1 / (t * Math.cbrt(((Math.sin(k_m) * Math.tan(k_m)) * ((k_m / t) * (k_m / t)))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64((cbrt(l) ^ 2.0) * cbrt(2.0))
	tmp = 0.0
	if (k_m <= 4e-9)
		tmp = Float64(t_1 / Float64(k_m * Float64(cbrt(Float64(k_m * t)) + Float64(0.05555555555555555 * cbrt(Float64(t * (k_m ^ 7.0))))))) ^ 3.0;
	elseif (k_m <= 1.46e+158)
		tmp = Float64(t_1 / cbrt(Float64(Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))) / cos(k_m)))) ^ 3.0;
	else
		tmp = Float64(t_1 / Float64(t * cbrt(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64(k_m / t) * Float64(k_m / t)))))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 4e-9], N[Power[N[(t$95$1 / N[(k$95$m * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 1/3], $MachinePrecision] + N[(0.05555555555555555 * N[Power[N[(t * N[Power[k$95$m, 7.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 1.46e+158], N[Power[N[(t$95$1 / N[Power[N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(t$95$1 / N[(t * N[Power[N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.00000000000000025e-9

   1. Initial program 38.6%

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

   3. Simplified 43.9%

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

      1. add-cube-cbrt 43.9%

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

      2. pow3 43.9%

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

   6. Applied egg-rr 58.0%

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

      1. pow2 58.0%

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

      2. *-commutative 58.0%

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

      3. cbrt-prod 58.0%

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

      4. cbrt-prod 70.8%

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

      5. unpow2 70.8%

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

   8. Applied egg-rr 70.8%

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

   9. Taylor expanded in k around 0 83.4%

      \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\color{blue}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{{k}^{7} \cdot t}\right)}}\right)}^{3} \]

   if 4.00000000000000025e-9 < k < 1.4599999999999999e158

   1. Initial program 25.2%

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

   3. Simplified 36.6%

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

      1. add-cube-cbrt 36.5%

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

      2. pow3 36.5%

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

   6. Applied egg-rr 60.0%

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

      1. pow2 60.0%

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

      2. *-commutative 60.0%

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

      3. cbrt-prod 59.8%

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

      4. cbrt-prod 68.2%

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

      5. unpow2 68.2%

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

   8. Applied egg-rr 68.2%

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

   9. Taylor expanded in t around 0 89.9%

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

   if 1.4599999999999999e158 < k

   1. Initial program 29.3%

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

   3. Simplified 29.2%

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

      1. add-cube-cbrt 29.2%

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

      2. pow3 29.2%

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

   6. Applied egg-rr 47.0%

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

      1. pow2 47.0%

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

      2. *-commutative 47.0%

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

      3. cbrt-prod 47.0%

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

      4. cbrt-prod 63.4%

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

      5. unpow2 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. unpow2 63.4%

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

   10. Applied egg-rr 63.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-9}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k}^{7}}\right)}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.46 \cdot 10^{+158}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\sqrt[3]{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{t \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}\\ \mathbf{if}\;k\_m \leq 0.7:\\ \;\;\;\;{\left(\frac{t\_1}{k\_m \cdot \left(\sqrt[3]{k\_m \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k\_m}^{7}}\right)}\right)}^{3}\\ \mathbf{elif}\;k\_m \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\cos k\_m \cdot \frac{{\ell}^{2}}{{\sin k\_m}^{2}}\right) \cdot \left({k\_m}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_1}{t \cdot \sqrt[3]{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{k\_m}{t} \cdot \frac{k\_m}{t}\right)}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (pow (cbrt l) 2.0) (cbrt 2.0))))
   (if (<= k_m 0.7)
     (pow
      (/
       t_1
       (*
        k_m
        (+
         (cbrt (* k_m t))
         (* 0.05555555555555555 (cbrt (* t (pow k_m 7.0)))))))
      3.0)
     (if (<= k_m 2e+158)
       (*
        (* (cos k_m) (/ (pow l 2.0) (pow (sin k_m) 2.0)))
        (* (pow k_m -2.0) (/ 2.0 t)))
       (pow
        (/
         t_1
         (* t (cbrt (* (* (sin k_m) (tan k_m)) (* (/ k_m t) (/ k_m t))))))
        3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), 2.0) * cbrt(2.0);
	double tmp;
	if (k_m <= 0.7) {
		tmp = pow((t_1 / (k_m * (cbrt((k_m * t)) + (0.05555555555555555 * cbrt((t * pow(k_m, 7.0))))))), 3.0);
	} else if (k_m <= 2e+158) {
		tmp = (cos(k_m) * (pow(l, 2.0) / pow(sin(k_m), 2.0))) * (pow(k_m, -2.0) * (2.0 / t));
	} else {
		tmp = pow((t_1 / (t * cbrt(((sin(k_m) * tan(k_m)) * ((k_m / t) * (k_m / t)))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(2.0);
	double tmp;
	if (k_m <= 0.7) {
		tmp = Math.pow((t_1 / (k_m * (Math.cbrt((k_m * t)) + (0.05555555555555555 * Math.cbrt((t * Math.pow(k_m, 7.0))))))), 3.0);
	} else if (k_m <= 2e+158) {
		tmp = (Math.cos(k_m) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k_m), 2.0))) * (Math.pow(k_m, -2.0) * (2.0 / t));
	} else {
		tmp = Math.pow((t_1 / (t * Math.cbrt(((Math.sin(k_m) * Math.tan(k_m)) * ((k_m / t) * (k_m / t)))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64((cbrt(l) ^ 2.0) * cbrt(2.0))
	tmp = 0.0
	if (k_m <= 0.7)
		tmp = Float64(t_1 / Float64(k_m * Float64(cbrt(Float64(k_m * t)) + Float64(0.05555555555555555 * cbrt(Float64(t * (k_m ^ 7.0))))))) ^ 3.0;
	elseif (k_m <= 2e+158)
		tmp = Float64(Float64(cos(k_m) * Float64((l ^ 2.0) / (sin(k_m) ^ 2.0))) * Float64((k_m ^ -2.0) * Float64(2.0 / t)));
	else
		tmp = Float64(t_1 / Float64(t * cbrt(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64(k_m / t) * Float64(k_m / t)))))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 0.7], N[Power[N[(t$95$1 / N[(k$95$m * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 1/3], $MachinePrecision] + N[(0.05555555555555555 * N[Power[N[(t * N[Power[k$95$m, 7.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 2e+158], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$1 / N[(t * N[Power[N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.69999999999999996

   1. Initial program 38.0%

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

   3. Simplified 43.8%

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

      1. add-cube-cbrt 43.7%

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

      2. pow3 43.8%

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

   6. Applied egg-rr 58.2%

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

      1. pow2 58.2%

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

      2. *-commutative 58.2%

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

      3. cbrt-prod 58.2%

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

      4. cbrt-prod 70.7%

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

      5. unpow2 70.7%

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

   8. Applied egg-rr 70.7%

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

   9. Taylor expanded in k around 0 83.6%

      \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\color{blue}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{{k}^{7} \cdot t}\right)}}\right)}^{3} \]

   if 0.69999999999999996 < k < 1.99999999999999991e158

   1. Initial program 27.0%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in t around 0 86.1%

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

      1. associate-*r* 86.1%

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

      2. *-commutative 86.1%

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

   7. Simplified 86.1%

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

   8. Taylor expanded in k around inf 86.0%

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

      1. associate-*r/ 86.0%

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

      2. times-frac 84.2%

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

      3. *-commutative 84.2%

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

   10. Simplified 84.2%

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

       1. associate-*r/ 81.1%

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

       2. div-inv 81.1%

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

       3. pow-flip 81.1%

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

       4. metadata-eval 81.1%

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

       5. pow2 81.1%

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

       6. *-commutative 81.1%

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

       7. pow2 81.1%

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

   12. Applied egg-rr 81.1%

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

       1. *-commutative 81.1%

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

       2. *-commutative 81.1%

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

       3. times-frac 86.3%

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

       4. *-commutative 86.3%

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

       5. associate-*r/ 86.5%

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

       6. *-commutative 86.5%

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

       7. associate-/l* 86.5%

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

   14. Simplified 86.5%

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

   if 1.99999999999999991e158 < k

   1. Initial program 29.3%

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

   3. Simplified 29.2%

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

      1. add-cube-cbrt 29.2%

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

      2. pow3 29.2%

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

   6. Applied egg-rr 47.0%

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

      1. pow2 47.0%

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

      2. *-commutative 47.0%

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

      3. cbrt-prod 47.0%

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

      4. cbrt-prod 63.4%

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

      5. unpow2 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. unpow2 63.4%

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

   10. Applied egg-rr 63.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.7:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k}^{7}}\right)}\right)}^{3}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \cdot \left({k}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{t \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;\left(\cos k\_m \cdot \frac{{\ell}^{2}}{{\sin k\_m}^{2}}\right) \cdot \left({k\_m}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k\_m}{t}\right)}^{2}}}{{\left(t \cdot \left(\sqrt[3]{\sin k\_m \cdot \tan k\_m} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{k\_m \cdot \left(\sqrt[3]{k\_m \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k\_m}^{7}}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.4e-76)
   (*
    (* (cos k_m) (/ (pow l 2.0) (pow (sin k_m) 2.0)))
    (* (pow k_m -2.0) (/ 2.0 t)))
   (if (<= t 4.3e+228)
     (/
      (/ 2.0 (pow (/ k_m t) 2.0))
      (pow (* t (* (cbrt (* (sin k_m) (tan k_m))) (pow (cbrt l) -2.0))) 3.0))
     (pow
      (/
       (cbrt (* 2.0 (pow l 2.0)))
       (*
        k_m
        (+
         (cbrt (* k_m t))
         (* 0.05555555555555555 (cbrt (* t (pow k_m 7.0)))))))
      3.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.4e-76) {
		tmp = (cos(k_m) * (pow(l, 2.0) / pow(sin(k_m), 2.0))) * (pow(k_m, -2.0) * (2.0 / t));
	} else if (t <= 4.3e+228) {
		tmp = (2.0 / pow((k_m / t), 2.0)) / pow((t * (cbrt((sin(k_m) * tan(k_m))) * pow(cbrt(l), -2.0))), 3.0);
	} else {
		tmp = pow((cbrt((2.0 * pow(l, 2.0))) / (k_m * (cbrt((k_m * t)) + (0.05555555555555555 * cbrt((t * pow(k_m, 7.0))))))), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.4e-76) {
		tmp = (Math.cos(k_m) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k_m), 2.0))) * (Math.pow(k_m, -2.0) * (2.0 / t));
	} else if (t <= 4.3e+228) {
		tmp = (2.0 / Math.pow((k_m / t), 2.0)) / Math.pow((t * (Math.cbrt((Math.sin(k_m) * Math.tan(k_m))) * Math.pow(Math.cbrt(l), -2.0))), 3.0);
	} else {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow(l, 2.0))) / (k_m * (Math.cbrt((k_m * t)) + (0.05555555555555555 * Math.cbrt((t * Math.pow(k_m, 7.0))))))), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.4e-76)
		tmp = Float64(Float64(cos(k_m) * Float64((l ^ 2.0) / (sin(k_m) ^ 2.0))) * Float64((k_m ^ -2.0) * Float64(2.0 / t)));
	elseif (t <= 4.3e+228)
		tmp = Float64(Float64(2.0 / (Float64(k_m / t) ^ 2.0)) / (Float64(t * Float64(cbrt(Float64(sin(k_m) * tan(k_m))) * (cbrt(l) ^ -2.0))) ^ 3.0));
	else
		tmp = Float64(cbrt(Float64(2.0 * (l ^ 2.0))) / Float64(k_m * Float64(cbrt(Float64(k_m * t)) + Float64(0.05555555555555555 * cbrt(Float64(t * (k_m ^ 7.0))))))) ^ 3.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.4e-76], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.3e+228], N[(N[(2.0 / N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(k$95$m * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 1/3], $MachinePrecision] + N[(0.05555555555555555 * N[Power[N[(t * N[Power[k$95$m, 7.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.3999999999999999e-76

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in k around inf 71.9%

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

      1. associate-*r/ 71.9%

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

      2. times-frac 72.1%

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

      3. *-commutative 72.1%

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

   10. Simplified 72.1%

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

       1. associate-*r/ 71.4%

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

       2. div-inv 71.4%

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

       3. pow-flip 71.7%

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

       4. metadata-eval 71.7%

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

       5. pow2 71.7%

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

       6. *-commutative 71.7%

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

       7. pow2 71.7%

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

   12. Applied egg-rr 71.7%

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

       1. *-commutative 71.7%

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

       2. *-commutative 71.7%

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

       3. times-frac 73.5%

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

       4. *-commutative 73.5%

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

       5. associate-*r/ 73.5%

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

       6. *-commutative 73.5%

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

       7. associate-/l* 73.5%

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

   14. Simplified 73.5%

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

   if 3.3999999999999999e-76 < t < 4.30000000000000032e228

   1. Initial program 43.1%

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

      1. *-commutative 43.1%

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

      2. associate-/r* 43.1%

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

   3. Simplified 51.9%

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

      1. unpow3 51.9%

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

      2. times-frac 59.8%

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

      3. pow2 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. add-cube-cbrt 59.7%

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

      2. pow3 59.7%

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

   8. Applied egg-rr 78.5%

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

      1. associate-*l* 80.1%

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

      2. *-commutative 80.1%

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

   10. Simplified 80.1%

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

   if 4.30000000000000032e228 < t

   1. Initial program 6.7%

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

   3. Simplified 20.0%

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

      1. add-cube-cbrt 20.0%

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

      2. pow3 20.0%

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

   6. Applied egg-rr 47.3%

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

   7. Taylor expanded in k around 0 79.9%

      \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{\color{blue}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{{k}^{7} \cdot t}\right)}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;\left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \cdot \left({k}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{\left(t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\ell}^{2}}}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k}^{7}}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.7:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{k\_m \cdot \left(\sqrt[3]{k\_m \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k\_m}^{7}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos k\_m \cdot \frac{{\ell}^{2}}{{\sin k\_m}^{2}}\right) \cdot \left({k\_m}^{-2} \cdot \frac{2}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.7)
   (pow
    (/
     (* (pow (cbrt l) 2.0) (cbrt 2.0))
     (*
      k_m
      (+ (cbrt (* k_m t)) (* 0.05555555555555555 (cbrt (* t (pow k_m 7.0)))))))
    3.0)
   (*
    (* (cos k_m) (/ (pow l 2.0) (pow (sin k_m) 2.0)))
    (* (pow k_m -2.0) (/ 2.0 t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.7) {
		tmp = pow(((pow(cbrt(l), 2.0) * cbrt(2.0)) / (k_m * (cbrt((k_m * t)) + (0.05555555555555555 * cbrt((t * pow(k_m, 7.0))))))), 3.0);
	} else {
		tmp = (cos(k_m) * (pow(l, 2.0) / pow(sin(k_m), 2.0))) * (pow(k_m, -2.0) * (2.0 / t));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.7) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(2.0)) / (k_m * (Math.cbrt((k_m * t)) + (0.05555555555555555 * Math.cbrt((t * Math.pow(k_m, 7.0))))))), 3.0);
	} else {
		tmp = (Math.cos(k_m) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k_m), 2.0))) * (Math.pow(k_m, -2.0) * (2.0 / t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.7)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * cbrt(2.0)) / Float64(k_m * Float64(cbrt(Float64(k_m * t)) + Float64(0.05555555555555555 * cbrt(Float64(t * (k_m ^ 7.0))))))) ^ 3.0;
	else
		tmp = Float64(Float64(cos(k_m) * Float64((l ^ 2.0) / (sin(k_m) ^ 2.0))) * Float64((k_m ^ -2.0) * Float64(2.0 / t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.7], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 1/3], $MachinePrecision] + N[(0.05555555555555555 * N[Power[N[(t * N[Power[k$95$m, 7.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.69999999999999996

   1. Initial program 38.0%

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

   3. Simplified 43.8%

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

      1. add-cube-cbrt 43.7%

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

      2. pow3 43.8%

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

   6. Applied egg-rr 58.2%

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

      1. pow2 58.2%

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

      2. *-commutative 58.2%

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

      3. cbrt-prod 58.2%

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

      4. cbrt-prod 70.7%

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

      5. unpow2 70.7%

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

   8. Applied egg-rr 70.7%

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

   9. Taylor expanded in k around 0 83.6%

      \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\color{blue}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{{k}^{7} \cdot t}\right)}}\right)}^{3} \]

   if 0.69999999999999996 < k

   1. Initial program 27.8%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in t around 0 71.5%

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

      1. associate-*r* 71.5%

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

      2. *-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in k around inf 71.5%

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

      1. associate-*r/ 71.5%

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

      2. times-frac 70.3%

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

      3. *-commutative 70.3%

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

   10. Simplified 70.3%

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

       1. associate-*r/ 68.4%

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

       2. div-inv 68.4%

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

       3. pow-flip 68.4%

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

       4. metadata-eval 68.4%

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

       5. pow2 68.4%

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

       6. *-commutative 68.4%

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

       7. pow2 68.4%

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

   12. Applied egg-rr 68.4%

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

       1. *-commutative 68.4%

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

       2. *-commutative 68.4%

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

       3. times-frac 71.7%

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

       4. *-commutative 71.7%

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

       5. associate-*r/ 71.8%

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

       6. *-commutative 71.8%

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

       7. associate-/l* 71.8%

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

   14. Simplified 71.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.7:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{k \cdot \left(\sqrt[3]{k \cdot t} + 0.05555555555555555 \cdot \sqrt[3]{t \cdot {k}^{7}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \cdot \left({k}^{-2} \cdot \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-159}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t \cdot {k\_m}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos k\_m \cdot \frac{{\ell}^{2}}{{\sin k\_m}^{2}}\right) \cdot \left({k\_m}^{-2} \cdot \frac{2}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.2e-159)
   (pow (/ (* (pow (cbrt l) 2.0) (cbrt 2.0)) (cbrt (* t (pow k_m 4.0)))) 3.0)
   (*
    (* (cos k_m) (/ (pow l 2.0) (pow (sin k_m) 2.0)))
    (* (pow k_m -2.0) (/ 2.0 t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e-159) {
		tmp = pow(((pow(cbrt(l), 2.0) * cbrt(2.0)) / cbrt((t * pow(k_m, 4.0)))), 3.0);
	} else {
		tmp = (cos(k_m) * (pow(l, 2.0) / pow(sin(k_m), 2.0))) * (pow(k_m, -2.0) * (2.0 / t));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.2e-159) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(2.0)) / Math.cbrt((t * Math.pow(k_m, 4.0)))), 3.0);
	} else {
		tmp = (Math.cos(k_m) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k_m), 2.0))) * (Math.pow(k_m, -2.0) * (2.0 / t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.2e-159)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * cbrt(2.0)) / cbrt(Float64(t * (k_m ^ 4.0)))) ^ 3.0;
	else
		tmp = Float64(Float64(cos(k_m) * Float64((l ^ 2.0) / (sin(k_m) ^ 2.0))) * Float64((k_m ^ -2.0) * Float64(2.0 / t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.2e-159], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e-159

   1. Initial program 39.3%

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

   3. Simplified 45.8%

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

      1. add-cube-cbrt 45.8%

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

      2. pow3 45.8%

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

   6. Applied egg-rr 59.7%

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

      1. pow2 59.7%

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

      2. *-commutative 59.7%

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

      3. cbrt-prod 59.6%

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

      4. cbrt-prod 74.5%

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

      5. unpow2 74.5%

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

   8. Applied egg-rr 74.5%

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

   9. Taylor expanded in k around 0 70.8%

      \[\leadsto {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\color{blue}{\sqrt[3]{{k}^{4} \cdot t}}}\right)}^{3} \]

   if 6.2e-159 < k

   1. Initial program 29.6%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. associate-*r* 73.2%

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

      2. *-commutative 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in k around inf 73.1%

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

      1. associate-*r/ 73.2%

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

      2. times-frac 72.6%

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

      3. *-commutative 72.6%

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

   10. Simplified 72.6%

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

       1. associate-*r/ 71.3%

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

       2. div-inv 71.3%

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

       3. pow-flip 71.3%

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

       4. metadata-eval 71.3%

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

       5. pow2 71.3%

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

       6. *-commutative 71.3%

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

       7. pow2 71.3%

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

   12. Applied egg-rr 71.3%

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

       1. *-commutative 71.3%

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

       2. *-commutative 71.3%

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

       3. times-frac 73.4%

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

       4. *-commutative 73.4%

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

       5. associate-*r/ 73.5%

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

       6. *-commutative 73.5%

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

       7. associate-/l* 73.5%

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

   14. Simplified 73.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-159}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{2}}{\sqrt[3]{t \cdot {k}^{4}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \cdot \left({k}^{-2} \cdot \frac{2}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-76} \lor \neg \left(t \leq 3.2 \cdot 10^{+106}\right):\\ \;\;\;\;\left(\cos k\_m \cdot \frac{{\ell}^{2}}{{\sin k\_m}^{2}}\right) \cdot \left({k\_m}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (or (<= t 1.9e-76) (not (<= t 3.2e+106)))
   (*
    (* (cos k_m) (/ (pow l 2.0) (pow (sin k_m) 2.0)))
    (* (pow k_m -2.0) (/ 2.0 t)))
   (/
    (/ 2.0 (/ (/ k_m t) (/ t k_m)))
    (/ (/ (* (* (sin k_m) (tan 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 <= 1.9e-76) || !(t <= 3.2e+106)) {
		tmp = (cos(k_m) * (pow(l, 2.0) / pow(sin(k_m), 2.0))) * (pow(k_m, -2.0) * (2.0 / t));
	} else {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(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 <= 1.9d-76) .or. (.not. (t <= 3.2d+106))) then
        tmp = (cos(k_m) * ((l ** 2.0d0) / (sin(k_m) ** 2.0d0))) * ((k_m ** (-2.0d0)) * (2.0d0 / t))
    else
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(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 <= 1.9e-76) || !(t <= 3.2e+106)) {
		tmp = (Math.cos(k_m) * (Math.pow(l, 2.0) / Math.pow(Math.sin(k_m), 2.0))) * (Math.pow(k_m, -2.0) * (2.0 / t));
	} else {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(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 <= 1.9e-76) or not (t <= 3.2e+106):
		tmp = (math.cos(k_m) * (math.pow(l, 2.0) / math.pow(math.sin(k_m), 2.0))) * (math.pow(k_m, -2.0) * (2.0 / t))
	else:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(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 <= 1.9e-76) || !(t <= 3.2e+106))
		tmp = Float64(Float64(cos(k_m) * Float64((l ^ 2.0) / (sin(k_m) ^ 2.0))) * Float64((k_m ^ -2.0) * Float64(2.0 / t)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((t <= 1.9e-76) || ~((t <= 3.2e+106)))
		tmp = (cos(k_m) * ((l ^ 2.0) / (sin(k_m) ^ 2.0))) * ((k_m ^ -2.0) * (2.0 / t));
	else
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(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[Or[LessEqual[t, 1.9e-76], N[Not[LessEqual[t, 3.2e+106]], $MachinePrecision]], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9000000000000001e-76 or 3.1999999999999998e106 < t

   1. Initial program 31.1%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r* 72.6%

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

      2. *-commutative 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in k around inf 72.8%

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

      1. associate-*r/ 72.8%

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

      2. times-frac 73.5%

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

      3. *-commutative 73.5%

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

   10. Simplified 73.5%

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

       1. associate-*r/ 72.9%

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

       2. div-inv 72.9%

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

       3. pow-flip 73.2%

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

       4. metadata-eval 73.2%

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

       5. pow2 73.2%

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

       6. *-commutative 73.2%

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

       7. pow2 73.2%

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

   12. Applied egg-rr 73.2%

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

       1. *-commutative 73.2%

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

       2. *-commutative 73.2%

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

       3. times-frac 74.2%

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

       4. *-commutative 74.2%

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

       5. associate-*r/ 74.2%

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

       6. *-commutative 74.2%

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

       7. associate-/l* 74.2%

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

   14. Simplified 74.2%

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

   if 1.9000000000000001e-76 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-76} \lor \neg \left(t \leq 3.2 \cdot 10^{+106}\right):\\ \;\;\;\;\left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \cdot \left({k}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ \mathbf{if}\;t \leq 3.2 \cdot 10^{-77}:\\ \;\;\;\;\left(\cos k\_m \cdot \frac{{\ell}^{2}}{t\_1}\right) \cdot \left({k\_m}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \left({\ell}^{2} \cdot \frac{\cos k\_m}{t \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)))
   (if (<= t 3.2e-77)
     (* (* (cos k_m) (/ (pow l 2.0) t_1)) (* (pow k_m -2.0) (/ 2.0 t)))
     (if (<= t 3.2e+106)
       (/
        (/ 2.0 (/ (/ k_m t) (/ t k_m)))
        (/ (/ (* (* (sin k_m) (tan k_m)) (pow t 3.0)) l) l))
       (* (/ 2.0 (pow k_m 2.0)) (* (pow l 2.0) (/ (cos k_m) (* t t_1))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double tmp;
	if (t <= 3.2e-77) {
		tmp = (cos(k_m) * (pow(l, 2.0) / t_1)) * (pow(k_m, -2.0) * (2.0 / t));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) * (cos(k_m) / (t * t_1)));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = sin(k_m) ** 2.0d0
    if (t <= 3.2d-77) then
        tmp = (cos(k_m) * ((l ** 2.0d0) / t_1)) * ((k_m ** (-2.0d0)) * (2.0d0 / t))
    else if (t <= 3.2d+106) then
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((l ** 2.0d0) * (cos(k_m) / (t * t_1)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (t <= 3.2e-77) {
		tmp = (Math.cos(k_m) * (Math.pow(l, 2.0) / t_1)) * (Math.pow(k_m, -2.0) * (2.0 / t));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) * (Math.cos(k_m) / (t * t_1)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if t <= 3.2e-77:
		tmp = (math.cos(k_m) * (math.pow(l, 2.0) / t_1)) * (math.pow(k_m, -2.0) * (2.0 / t))
	elif t <= 3.2e+106:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(k_m)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) * (math.cos(k_m) / (t * t_1)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (t <= 3.2e-77)
		tmp = Float64(Float64(cos(k_m) * Float64((l ^ 2.0) / t_1)) * Float64((k_m ^ -2.0) * Float64(2.0 / t)));
	elseif (t <= 3.2e+106)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64((l ^ 2.0) * Float64(cos(k_m) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0;
	tmp = 0.0;
	if (t <= 3.2e-77)
		tmp = (cos(k_m) * ((l ^ 2.0) / t_1)) * ((k_m ^ -2.0) * (2.0 / t));
	elseif (t <= 3.2e+106)
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (2.0 / (k_m ^ 2.0)) * ((l ^ 2.0) * (cos(k_m) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 3.2e-77], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+106], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.2e-77

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in k around inf 71.9%

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

      1. associate-*r/ 71.9%

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

      2. times-frac 72.1%

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

      3. *-commutative 72.1%

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

   10. Simplified 72.1%

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

       1. associate-*r/ 71.4%

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

       2. div-inv 71.4%

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

       3. pow-flip 71.7%

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

       4. metadata-eval 71.7%

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

       5. pow2 71.7%

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

       6. *-commutative 71.7%

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

       7. pow2 71.7%

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

   12. Applied egg-rr 71.7%

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

       1. *-commutative 71.7%

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

       2. *-commutative 71.7%

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

       3. times-frac 73.5%

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

       4. *-commutative 73.5%

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

       5. associate-*r/ 73.5%

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

       6. *-commutative 73.5%

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

       7. associate-/l* 73.5%

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

   14. Simplified 73.5%

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

   if 3.2e-77 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 3.1999999999999998e106 < t

   1. Initial program 12.8%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in t around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in k around inf 77.3%

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

      1. associate-*r/ 77.3%

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

      2. times-frac 79.7%

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

      3. *-commutative 79.7%

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

   10. Simplified 79.7%

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

   11. Taylor expanded in k around inf 79.7%

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

       1. associate-/l* 79.7%

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

   13. Simplified 79.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-77}:\\ \;\;\;\;\left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2}}\right) \cdot \left({k}^{-2} \cdot \frac{2}{t}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\_m\right)}{t \cdot {\left(k\_m \cdot \sin k\_m\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k\_m \cdot {\ell}^{2}}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.7e-76)
   (/ (* (pow l 2.0) (* 2.0 (cos k_m))) (* t (pow (* k_m (sin k_m)) 2.0)))
   (if (<= t 3.2e+106)
     (/
      (/ 2.0 (/ (/ k_m t) (/ t k_m)))
      (/ (/ (* (* (sin k_m) (tan k_m)) (pow t 3.0)) l) l))
     (*
      2.0
      (/
       (* (cos k_m) (pow l 2.0))
       (* (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 (t <= 4.7e-76) {
		tmp = (pow(l, 2.0) * (2.0 * cos(k_m))) / (t * pow((k_m * sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = 2.0 * ((cos(k_m) * pow(l, 2.0)) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 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 (t <= 4.7d-76) then
        tmp = ((l ** 2.0d0) * (2.0d0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ** 2.0d0))
    else if (t <= 3.2d+106) then
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = 2.0d0 * ((cos(k_m) * (l ** 2.0d0)) / ((k_m ** 2.0d0) * (t * (sin(k_m) ** 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 (t <= 4.7e-76) {
		tmp = (Math.pow(l, 2.0) * (2.0 * Math.cos(k_m))) / (t * Math.pow((k_m * Math.sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * Math.pow(l, 2.0)) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.7e-76:
		tmp = (math.pow(l, 2.0) * (2.0 * math.cos(k_m))) / (t * math.pow((k_m * math.sin(k_m)), 2.0))
	elif t <= 3.2e+106:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(k_m)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = 2.0 * ((math.cos(k_m) * math.pow(l, 2.0)) / (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 (t <= 4.7e-76)
		tmp = Float64(Float64((l ^ 2.0) * Float64(2.0 * cos(k_m))) / Float64(t * (Float64(k_m * sin(k_m)) ^ 2.0)));
	elseif (t <= 3.2e+106)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * (l ^ 2.0)) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 4.7e-76)
		tmp = ((l ^ 2.0) * (2.0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ^ 2.0));
	elseif (t <= 3.2e+106)
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l);
	else
		tmp = 2.0 * ((cos(k_m) * (l ^ 2.0)) / ((k_m ^ 2.0) * (t * (sin(k_m) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.7e-76], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+106], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $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 3 regimes

2. if t < 4.7000000000000002e-76

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. associate-*r* 69.3%

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

      3. pow-prod-down 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. associate-*l/ 69.4%

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

       2. pow2 69.4%

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

       3. *-commutative 69.4%

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

       4. *-commutative 69.4%

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

   11. Applied egg-rr 69.4%

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

   if 4.7000000000000002e-76 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 3.1999999999999998e106 < t

   1. Initial program 12.8%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in t around 0 77.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\right)}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\_m\right)}{t \cdot {\left(k\_m \cdot \sin k\_m\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t \cdot {\sin k\_m}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6e-76)
   (/ (* (pow l 2.0) (* 2.0 (cos k_m))) (* t (pow (* k_m (sin k_m)) 2.0)))
   (if (<= t 3.2e+106)
     (/
      (/ 2.0 (/ (/ k_m t) (/ t k_m)))
      (/ (/ (* (* (sin k_m) (tan k_m)) (pow t 3.0)) l) l))
     (*
      (* (/ 2.0 (pow k_m 2.0)) (/ (cos k_m) (* t (pow (sin k_m) 2.0))))
      (* l l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6e-76) {
		tmp = (pow(l, 2.0) * (2.0 * cos(k_m))) / (t * pow((k_m * sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = ((2.0 / pow(k_m, 2.0)) * (cos(k_m) / (t * pow(sin(k_m), 2.0)))) * (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 <= 6d-76) then
        tmp = ((l ** 2.0d0) * (2.0d0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ** 2.0d0))
    else if (t <= 3.2d+106) then
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = ((2.0d0 / (k_m ** 2.0d0)) * (cos(k_m) / (t * (sin(k_m) ** 2.0d0)))) * (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 <= 6e-76) {
		tmp = (Math.pow(l, 2.0) * (2.0 * Math.cos(k_m))) / (t * Math.pow((k_m * Math.sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = ((2.0 / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t * Math.pow(Math.sin(k_m), 2.0)))) * (l * l);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 6e-76:
		tmp = (math.pow(l, 2.0) * (2.0 * math.cos(k_m))) / (t * math.pow((k_m * math.sin(k_m)), 2.0))
	elif t <= 3.2e+106:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(k_m)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = ((2.0 / math.pow(k_m, 2.0)) * (math.cos(k_m) / (t * math.pow(math.sin(k_m), 2.0)))) * (l * l)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6e-76)
		tmp = Float64(Float64((l ^ 2.0) * Float64(2.0 * cos(k_m))) / Float64(t * (Float64(k_m * sin(k_m)) ^ 2.0)));
	elseif (t <= 3.2e+106)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t * (sin(k_m) ^ 2.0)))) * 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 <= 6e-76)
		tmp = ((l ^ 2.0) * (2.0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ^ 2.0));
	elseif (t <= 3.2e+106)
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l);
	else
		tmp = ((2.0 / (k_m ^ 2.0)) * (cos(k_m) / (t * (sin(k_m) ^ 2.0)))) * (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6e-76], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+106], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.00000000000000048e-76

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. associate-*r* 69.3%

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

      3. pow-prod-down 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. associate-*l/ 69.4%

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

       2. pow2 69.4%

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

       3. *-commutative 69.4%

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

       4. *-commutative 69.4%

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

   11. Applied egg-rr 69.4%

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

   if 6.00000000000000048e-76 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 3.1999999999999998e106 < t

   1. Initial program 12.8%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in t around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in k around inf 77.2%

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

      1. associate-*r/ 77.2%

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

      2. *-commutative 77.2%

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

      3. *-commutative 77.2%

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

      4. times-frac 77.3%

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

   10. Simplified 77.3%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\right)}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\_m\right)}{t \cdot {\left(k\_m \cdot \sin k\_m\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k\_m}{{k\_m}^{2} \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6.8e-76)
   (/ (* (pow l 2.0) (* 2.0 (cos k_m))) (* t (pow (* k_m (sin k_m)) 2.0)))
   (if (<= t 3.2e+106)
     (/
      (/ 2.0 (/ (/ k_m t) (/ t k_m)))
      (/ (/ (* (* (sin k_m) (tan k_m)) (pow t 3.0)) l) l))
     (*
      (* l 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 (t <= 6.8e-76) {
		tmp = (pow(l, 2.0) * (2.0 * cos(k_m))) / (t * pow((k_m * sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 * (cos(k_m) / (pow(k_m, 2.0) * (t * pow(sin(k_m), 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 (t <= 6.8d-76) then
        tmp = ((l ** 2.0d0) * (2.0d0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ** 2.0d0))
    else if (t <= 3.2d+106) then
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (l * l) * (2.0d0 * (cos(k_m) / ((k_m ** 2.0d0) * (t * (sin(k_m) ** 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 (t <= 6.8e-76) {
		tmp = (Math.pow(l, 2.0) * (2.0 * Math.cos(k_m))) / (t * Math.pow((k_m * Math.sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 * (Math.cos(k_m) / (Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 6.8e-76:
		tmp = (math.pow(l, 2.0) * (2.0 * math.cos(k_m))) / (t * math.pow((k_m * math.sin(k_m)), 2.0))
	elif t <= 3.2e+106:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(k_m)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (l * 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 (t <= 6.8e-76)
		tmp = Float64(Float64((l ^ 2.0) * Float64(2.0 * cos(k_m))) / Float64(t * (Float64(k_m * sin(k_m)) ^ 2.0)));
	elseif (t <= 3.2e+106)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(cos(k_m) / Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 6.8e-76)
		tmp = ((l ^ 2.0) * (2.0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ^ 2.0));
	elseif (t <= 3.2e+106)
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (l * l) * (2.0 * (cos(k_m) / ((k_m ^ 2.0) * (t * (sin(k_m) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6.8e-76], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+106], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $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]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.7999999999999998e-76

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. associate-*r* 69.3%

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

      3. pow-prod-down 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. associate-*l/ 69.4%

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

       2. pow2 69.4%

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

       3. *-commutative 69.4%

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

       4. *-commutative 69.4%

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

   11. Applied egg-rr 69.4%

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

   if 6.7999999999999998e-76 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 3.1999999999999998e106 < t

   1. Initial program 12.8%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in t around 0 77.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\right)}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\_m\right)}{t \cdot {\left(k\_m \cdot \sin k\_m\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 9e-76)
   (/ (* (pow l 2.0) (* 2.0 (cos k_m))) (* t (pow (* k_m (sin k_m)) 2.0)))
   (if (<= t 3.2e+106)
     (/
      (/ 2.0 (/ (/ k_m t) (/ t k_m)))
      (/ (/ (* (* (sin k_m) (tan k_m)) (pow t 3.0)) l) l))
     (* (/ 2.0 (pow k_m 2.0)) (/ (pow l 2.0) (* t (pow k_m 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9e-76) {
		tmp = (pow(l, 2.0) * (2.0 * cos(k_m))) / (t * pow((k_m * sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) / (t * pow(k_m, 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 (t <= 9d-76) then
        tmp = ((l ** 2.0d0) * (2.0d0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ** 2.0d0))
    else if (t <= 3.2d+106) then
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((l ** 2.0d0) / (t * (k_m ** 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 (t <= 9e-76) {
		tmp = (Math.pow(l, 2.0) * (2.0 * Math.cos(k_m))) / (t * Math.pow((k_m * Math.sin(k_m)), 2.0));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 9e-76:
		tmp = (math.pow(l, 2.0) * (2.0 * math.cos(k_m))) / (t * math.pow((k_m * math.sin(k_m)), 2.0))
	elif t <= 3.2e+106:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(k_m)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) / (t * math.pow(k_m, 2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 9e-76)
		tmp = Float64(Float64((l ^ 2.0) * Float64(2.0 * cos(k_m))) / Float64(t * (Float64(k_m * sin(k_m)) ^ 2.0)));
	elseif (t <= 3.2e+106)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * (k_m ^ 2.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 9e-76)
		tmp = ((l ^ 2.0) * (2.0 * cos(k_m))) / (t * ((k_m * sin(k_m)) ^ 2.0));
	elseif (t <= 3.2e+106)
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (2.0 / (k_m ^ 2.0)) * ((l ^ 2.0) / (t * (k_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 9e-76], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+106], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.0000000000000001e-76

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. associate-*r* 69.3%

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

      3. pow-prod-down 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. associate-*l/ 69.4%

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

       2. pow2 69.4%

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

       3. *-commutative 69.4%

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

       4. *-commutative 69.4%

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

   11. Applied egg-rr 69.4%

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

   if 9.0000000000000001e-76 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 3.1999999999999998e106 < t

   1. Initial program 12.8%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in t around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in k around inf 77.3%

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

      1. associate-*r/ 77.3%

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

      2. times-frac 79.7%

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

      3. *-commutative 79.7%

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

   10. Simplified 79.7%

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

   11. Taylor expanded in k around 0 74.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-76}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \left(2 \cdot \cos k\right)}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{1}{{\left(k\_m \cdot \sin k\_m\right)}^{2} \cdot \frac{t}{\cos k\_m}}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k\_m}{t}}{\frac{t}{k\_m}}}}{\frac{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5e-76)
   (* (* l l) (* 2.0 (/ 1.0 (* (pow (* k_m (sin k_m)) 2.0) (/ t (cos k_m))))))
   (if (<= t 3.2e+106)
     (/
      (/ 2.0 (/ (/ k_m t) (/ t k_m)))
      (/ (/ (* (* (sin k_m) (tan k_m)) (pow t 3.0)) l) l))
     (* (/ 2.0 (pow k_m 2.0)) (/ (pow l 2.0) (* t (pow k_m 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5e-76) {
		tmp = (l * l) * (2.0 * (1.0 / (pow((k_m * sin(k_m)), 2.0) * (t / cos(k_m)))));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) / (t * pow(k_m, 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 (t <= 5d-76) then
        tmp = (l * l) * (2.0d0 * (1.0d0 / (((k_m * sin(k_m)) ** 2.0d0) * (t / cos(k_m)))))
    else if (t <= 3.2d+106) then
        tmp = (2.0d0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((l ** 2.0d0) / (t * (k_m ** 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 (t <= 5e-76) {
		tmp = (l * l) * (2.0 * (1.0 / (Math.pow((k_m * Math.sin(k_m)), 2.0) * (t / Math.cos(k_m)))));
	} else if (t <= 3.2e+106) {
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((Math.sin(k_m) * Math.tan(k_m)) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 2.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 5e-76:
		tmp = (l * l) * (2.0 * (1.0 / (math.pow((k_m * math.sin(k_m)), 2.0) * (t / math.cos(k_m)))))
	elif t <= 3.2e+106:
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((math.sin(k_m) * math.tan(k_m)) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) / (t * math.pow(k_m, 2.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5e-76)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(1.0 / Float64((Float64(k_m * sin(k_m)) ^ 2.0) * Float64(t / cos(k_m))))));
	elseif (t <= 3.2e+106)
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m / t) / Float64(t / k_m))) / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * (k_m ^ 2.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 5e-76)
		tmp = (l * l) * (2.0 * (1.0 / (((k_m * sin(k_m)) ^ 2.0) * (t / cos(k_m)))));
	elseif (t <= 3.2e+106)
		tmp = (2.0 / ((k_m / t) / (t / k_m))) / ((((sin(k_m) * tan(k_m)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (2.0 / (k_m ^ 2.0)) * ((l ^ 2.0) / (t * (k_m ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5e-76], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(1.0 / N[(N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e+106], N[(N[(2.0 / N[(N[(k$95$m / t), $MachinePrecision] / N[(t / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.9999999999999998e-76

   1. Initial program 35.1%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

   7. Simplified 71.6%

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

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. associate-*r* 69.3%

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

      4. pow-prod-down 69.4%

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

   9. Applied egg-rr 69.4%

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

       1. unpow-1 69.4%

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

       2. associate-/l* 69.4%

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

       3. *-commutative 69.4%

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

   11. Simplified 69.4%

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

   if 4.9999999999999998e-76 < t < 3.1999999999999998e106

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-/r* 58.9%

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

   3. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. clear-num 65.5%

         \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \color{blue}{\frac{1}{\frac{t}{k}}} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. un-div-inv 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.4%

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

      2. associate-/r* 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 3.1999999999999998e106 < t

   1. Initial program 12.8%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in t around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in k around inf 77.3%

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

      1. associate-*r/ 77.3%

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

      2. times-frac 79.7%

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

      3. *-commutative 79.7%

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

   10. Simplified 79.7%

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

   11. Taylor expanded in k around 0 74.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{1}{{\left(k \cdot \sin k\right)}^{2} \cdot \frac{t}{\cos k}}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{\left(k\_m \cdot \sin k\_m\right)}^{2}} \cdot \frac{\cos k\_m}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-15)
   (* (/ 2.0 (pow k_m 2.0)) (/ (pow l 2.0) (* t (pow k_m 2.0))))
   (* (* l l) (* (/ 2.0 (pow (* k_m (sin k_m)) 2.0)) (/ (cos k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-15) {
		tmp = (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) / (t * pow(k_m, 2.0)));
	} else {
		tmp = (l * l) * ((2.0 / pow((k_m * sin(k_m)), 2.0)) * (cos(k_m) / t));
	}
	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 <= 9.2d-15) then
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((l ** 2.0d0) / (t * (k_m ** 2.0d0)))
    else
        tmp = (l * l) * ((2.0d0 / ((k_m * sin(k_m)) ** 2.0d0)) * (cos(k_m) / t))
    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 <= 9.2e-15) {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 2.0)));
	} else {
		tmp = (l * l) * ((2.0 / Math.pow((k_m * Math.sin(k_m)), 2.0)) * (Math.cos(k_m) / t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9.2e-15:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) / (t * math.pow(k_m, 2.0)))
	else:
		tmp = (l * l) * ((2.0 / math.pow((k_m * math.sin(k_m)), 2.0)) * (math.cos(k_m) / t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-15)
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * (k_m ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (Float64(k_m * sin(k_m)) ^ 2.0)) * Float64(cos(k_m) / t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9.2e-15)
		tmp = (2.0 / (k_m ^ 2.0)) * ((l ^ 2.0) / (t * (k_m ^ 2.0)));
	else
		tmp = (l * l) * ((2.0 / ((k_m * sin(k_m)) ^ 2.0)) * (cos(k_m) / t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-15], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.19999999999999961e-15

   1. Initial program 38.6%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in t around 0 71.5%

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

      1. associate-*r* 71.5%

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

      2. *-commutative 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in k around inf 71.9%

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

      1. associate-*r/ 71.9%

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

      2. times-frac 73.1%

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

      3. *-commutative 73.1%

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

   10. Simplified 73.1%

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

   11. Taylor expanded in k around 0 67.8%

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

   if 9.19999999999999961e-15 < k

   1. Initial program 27.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r* 72.6%

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

      2. *-commutative 72.6%

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

   7. Simplified 72.6%

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

      1. associate-*r/ 72.6%

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

      2. associate-*r* 72.6%

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

      3. pow-prod-down 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. times-frac 72.7%

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

       2. *-commutative 72.7%

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

   11. Simplified 72.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) / (t * pow(k_m, 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 ** 2.0d0)) * ((l ** 2.0d0) / (t * (k_m ** 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, 2.0)) * (Math.pow(l, 2.0) / (t * Math.pow(k_m, 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r* 71.8%

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

   2. *-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in k around inf 72.0%

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

   1. associate-*r/ 72.0%

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

   2. times-frac 72.6%

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

   3. *-commutative 72.6%

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

10. Simplified 72.6%

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

11. Taylor expanded in k around 0 63.6%

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

12. Final simplification 63.6%

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

Alternative 16: 63.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) * (2.0 / (t * pow((k_m * sin(k_m)), 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 = (l * l) * (2.0d0 / (t * ((k_m * sin(k_m)) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r* 71.8%

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

   2. *-commutative 71.8%

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

7. Simplified 71.8%

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

   1. associate-*r/ 71.8%

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

   2. associate-*r* 68.7%

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

   3. pow-prod-down 68.8%

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

9. Applied egg-rr 68.8%

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

10. Taylor expanded in k around 0 59.9%

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

11. Final simplification 59.9%

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

Alternative 17: 63.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) * (2.0 * (cos(k_m) / (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 = (l * l) * (2.0d0 * (cos(k_m) / (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 (l * l) * (2.0 * (Math.cos(k_m) / (t * Math.pow(k_m, 4.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in t around 0 71.8%

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

   1. associate-*r* 71.8%

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

   2. *-commutative 71.8%

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

7. Simplified 71.8%

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

8. Taylor expanded in k around 0 57.9%

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

9. Final simplification 57.9%

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

Alternative 18: 62.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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 = (l * l) * (2.0d0 * ((k_m ** (-4.0d0)) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.4%

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

3. Simplified 41.3%

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

5. Taylor expanded in k around 0 57.5%

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

   1. *-commutative 57.5%

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

   2. associate-/r* 57.5%

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

7. Simplified 57.5%

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

   1. div-inv 57.5%

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

   2. pow-flip 57.5%

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

   3. metadata-eval 57.5%

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

9. Applied egg-rr 57.5%

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

    1. associate-*l/ 57.5%

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

    2. associate-/l* 57.5%

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

11. Simplified 57.5%

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

12. Final simplification 57.5%

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

Reproduce

herbie shell --seed 2024139 
(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))))