Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.1% → 97.8%

Time: 13.8s

Alternatives: 19

Speedup: 9.6×

• 39.1% 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: 34.1% 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: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m}{\cos k\_m}}{\ell} \cdot \left({\sin k\_m}^{2} \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e-60)
   (/
    2.0
    (/
     (*
      (*
       (*
        (fma
         (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
         (* k_m k_m)
         t)
        k_m)
       k_m)
      (/ k_m l))
     (/ (* (cos k_m) l) k_m)))
   (/
    2.0
    (* (/ (/ k_m (cos k_m)) l) (* (pow (sin k_m) 2.0) (* t (/ k_m l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e-60) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
	} else {
		tmp = 2.0 / (((k_m / cos(k_m)) / l) * (pow(sin(k_m), 2.0) * (t * (k_m / l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / cos(k_m)) / l) * Float64((sin(k_m) ^ 2.0) * Float64(t * Float64(k_m / l)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e-60], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-60

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.25e-60 < k

   1. Initial program 28.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 97.7%

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

Alternative 2: 84.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-221}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{2}{{\left(\sin k\_m \cdot k\_m\right)}^{2} \cdot \frac{t}{\left(\ell \cdot \cos k\_m\right) \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \frac{k\_m}{t}\right) \cdot \frac{k\_m}{t}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= (* l l) 2e-221)
   (/
    2.0
    (/
     (*
      (*
       (*
        (fma
         (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
         (* k_m k_m)
         t)
        k_m)
       k_m)
      (/ k_m l))
     (/ (* (cos k_m) l) k_m)))
   (if (<= (* l l) 5e+306)
     (/ 2.0 (* (pow (* (sin k_m) k_m) 2.0) (/ t (* (* l (cos k_m)) l))))
     (/
      2.0
      (*
       (/ t l)
       (*
        (* (* (* (tan k_m) (sin k_m)) (* (/ t l) t)) (/ k_m t))
        (/ k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((l * l) <= 2e-221) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
	} else if ((l * l) <= 5e+306) {
		tmp = 2.0 / (pow((sin(k_m) * k_m), 2.0) * (t / ((l * cos(k_m)) * l)));
	} else {
		tmp = 2.0 / ((t / l) * ((((tan(k_m) * sin(k_m)) * ((t / l) * t)) * (k_m / t)) * (k_m / t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 2e-221)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)));
	elseif (Float64(l * l) <= 5e+306)
		tmp = Float64(2.0 / Float64((Float64(sin(k_m) * k_m) ^ 2.0) * Float64(t / Float64(Float64(l * cos(k_m)) * l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(Float64(t / l) * t)) * Float64(k_m / t)) * Float64(k_m / t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e-221], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+306], N[(2.0 / N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(t / N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.00000000000000003e-221

   1. Initial program 24.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 84.6%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 91.5%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 2.00000000000000003e-221 < (*.f64 l l) < 4.99999999999999993e306

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 87.2%

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

   11. Applied rewrites 87.6%

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

   if 4.99999999999999993e306 < (*.f64 l l)

   1. Initial program 37.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 57.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 63.5

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

Alternative 3: 89.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 3 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{t\_1}{k\_m}}}\\ \mathbf{elif}\;k\_m \leq 1.5 \cdot 10^{+178}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\ell} \cdot \frac{t}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k\_m}^{2} \cdot \left(\left(k\_m \cdot t\right) \cdot \frac{k\_m}{t\_1 \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 3e-66)
     (/
      2.0
      (/
       (*
        (*
         (*
          (fma
           (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
           (* k_m k_m)
           t)
          k_m)
         k_m)
        (/ k_m l))
       (/ t_1 k_m)))
     (if (<= k_m 1.5e+178)
       (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) l) (/ t (* l (cos k_m)))))
       (/ 2.0 (* (pow (sin k_m) 2.0) (* (* k_m t) (/ k_m (* t_1 l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 3e-66) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / (t_1 / k_m));
	} else if (k_m <= 1.5e+178) {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / l) * (t / (l * cos(k_m))));
	} else {
		tmp = 2.0 / (pow(sin(k_m), 2.0) * ((k_m * t) * (k_m / (t_1 * l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 3e-66)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(t_1 / k_m)));
	elseif (k_m <= 1.5e+178)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / l) * Float64(t / Float64(l * cos(k_m)))));
	else
		tmp = Float64(2.0 / Float64((sin(k_m) ^ 2.0) * Float64(Float64(k_m * t) * Float64(k_m / Float64(t_1 * l)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 3e-66], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.5e+178], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m / N[(t$95$1 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.0000000000000002e-66

   1. Initial program 38.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 94.1

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.9%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 3.0000000000000002e-66 < k < 1.50000000000000008e178

   1. Initial program 21.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 87.7

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

   5. Applied rewrites 87.7%

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

   7. Applied rewrites 97.5%

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

   9. Applied rewrites 73.8%

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

   11. Applied rewrites 93.3%

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

   if 1.50000000000000008e178 < k

   1. Initial program 40.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 97.1%

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

   9. Applied rewrites 83.1%

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

Alternative 4: 88.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{t\_2}{k\_m}}}\\ \mathbf{elif}\;k\_m \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t\_1} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(k\_m \cdot t\right) \cdot \frac{k\_m}{t\_2 \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* (cos k_m) l)))
   (if (<= k_m 1.05e-59)
     (/
      2.0
      (/
       (*
        (*
         (*
          (fma
           (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
           (* k_m k_m)
           t)
          k_m)
         k_m)
        (/ k_m l))
       (/ t_2 k_m)))
     (if (<= k_m 2e+118)
       (* (/ (* 2.0 (cos k_m)) (* (* k_m k_m) t_1)) (* l (/ l t)))
       (/ 2.0 (* t_1 (* (* k_m t) (/ k_m (* t_2 l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = cos(k_m) * l;
	double tmp;
	if (k_m <= 1.05e-59) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / (t_2 / k_m));
	} else if (k_m <= 2e+118) {
		tmp = ((2.0 * cos(k_m)) / ((k_m * k_m) * t_1)) * (l * (l / t));
	} else {
		tmp = 2.0 / (t_1 * ((k_m * t) * (k_m / (t_2 * l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 1.05e-59)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(t_2 / k_m)));
	elseif (k_m <= 2e+118)
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(k_m * k_m) * t_1)) * Float64(l * Float64(l / t)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k_m * t) * Float64(k_m / Float64(t_2 * l)))));
	end
	return 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]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.05e-59], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2e+118], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m / N[(t$95$2 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.04999999999999998e-59

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.04999999999999998e-59 < k < 1.99999999999999993e118

   1. Initial program 27.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 88.5

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

   5. Applied rewrites 88.5%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around 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/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

   10. Applied rewrites 94.3%

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

   if 1.99999999999999993e118 < k

   1. Initial program 30.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 96.2%

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

   9. Applied rewrites 79.6%

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

Alternative 5: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{t\_2}{k\_m}}}\\ \mathbf{elif}\;k\_m \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t\_1} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{\left(t\_1 \cdot t\right) \cdot k\_m}{t\_2 \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* (cos k_m) l)))
   (if (<= k_m 1.05e-59)
     (/
      2.0
      (/
       (*
        (*
         (*
          (fma
           (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
           (* k_m k_m)
           t)
          k_m)
         k_m)
        (/ k_m l))
       (/ t_2 k_m)))
     (if (<= k_m 2e+118)
       (* (/ (* 2.0 (cos k_m)) (* (* k_m k_m) t_1)) (* l (/ l t)))
       (/ 2.0 (* k_m (/ (* (* t_1 t) k_m) (* t_2 l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = cos(k_m) * l;
	double tmp;
	if (k_m <= 1.05e-59) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / (t_2 / k_m));
	} else if (k_m <= 2e+118) {
		tmp = ((2.0 * cos(k_m)) / ((k_m * k_m) * t_1)) * (l * (l / t));
	} else {
		tmp = 2.0 / (k_m * (((t_1 * t) * k_m) / (t_2 * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 1.05e-59)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(t_2 / k_m)));
	elseif (k_m <= 2e+118)
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(k_m * k_m) * t_1)) * Float64(l * Float64(l / t)));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(Float64(t_1 * t) * k_m) / Float64(t_2 * l))));
	end
	return 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]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.05e-59], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2e+118], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(N[(t$95$1 * t), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(t$95$2 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.04999999999999998e-59

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.04999999999999998e-59 < k < 1.99999999999999993e118

   1. Initial program 27.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 88.5

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

   5. Applied rewrites 88.5%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around 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/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

   10. Applied rewrites 94.3%

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

   if 1.99999999999999993e118 < k

   1. Initial program 30.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 96.2%

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

   9. Applied rewrites 77.3%

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

Alternative 6: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\sin k\_m}^{2}\\ t_2 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{t\_2}{k\_m}}}\\ \mathbf{elif}\;k\_m \leq 1.85 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t\_1} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\left(t\_1 \cdot t\right) \cdot \frac{k\_m}{t\_2 \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (sin k_m) 2.0)) (t_2 (* (cos k_m) l)))
   (if (<= k_m 1.05e-59)
     (/
      2.0
      (/
       (*
        (*
         (*
          (fma
           (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
           (* k_m k_m)
           t)
          k_m)
         k_m)
        (/ k_m l))
       (/ t_2 k_m)))
     (if (<= k_m 1.85e+81)
       (* (/ (* 2.0 (cos k_m)) (* (* k_m k_m) t_1)) (* l (/ l t)))
       (/ 2.0 (* k_m (* (* t_1 t) (/ k_m (* t_2 l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(sin(k_m), 2.0);
	double t_2 = cos(k_m) * l;
	double tmp;
	if (k_m <= 1.05e-59) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / (t_2 / k_m));
	} else if (k_m <= 1.85e+81) {
		tmp = ((2.0 * cos(k_m)) / ((k_m * k_m) * t_1)) * (l * (l / t));
	} else {
		tmp = 2.0 / (k_m * ((t_1 * t) * (k_m / (t_2 * l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = sin(k_m) ^ 2.0
	t_2 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 1.05e-59)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(t_2 / k_m)));
	elseif (k_m <= 1.85e+81)
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(k_m * k_m) * t_1)) * Float64(l * Float64(l / t)));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(t_1 * t) * Float64(k_m / Float64(t_2 * l)))));
	end
	return 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]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.05e-59], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.85e+81], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(t$95$1 * t), $MachinePrecision] * N[(k$95$m / N[(t$95$2 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.04999999999999998e-59

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.04999999999999998e-59 < k < 1.85e81

   1. Initial program 27.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around 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/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

   10. Applied rewrites 93.6%

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

   if 1.85e81 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 90.3

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 96.5%

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

   9. Applied rewrites 79.4%

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

Alternative 7: 80.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}}\\ \mathbf{elif}\;\ell \leq 1.02 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\ell \cdot \ell}{{\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \frac{k\_m}{t}\right) \cdot \frac{k\_m}{t}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 9e-161)
   (/
    2.0
    (/
     (*
      (*
       (*
        (fma
         (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
         (* k_m k_m)
         t)
        k_m)
       k_m)
      (/ k_m l))
     (/ (* (cos k_m) l) k_m)))
   (if (<= l 1.02e+154)
     (*
      (/ (* 2.0 (cos k_m)) (* (* k_m k_m) t))
      (/ (* l l) (pow (sin k_m) 2.0)))
     (/
      2.0
      (*
       (/ t l)
       (*
        (* (* (* (tan k_m) (sin k_m)) (* (/ t l) t)) (/ k_m t))
        (/ k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 9e-161) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
	} else if (l <= 1.02e+154) {
		tmp = ((2.0 * cos(k_m)) / ((k_m * k_m) * t)) * ((l * l) / pow(sin(k_m), 2.0));
	} else {
		tmp = 2.0 / ((t / l) * ((((tan(k_m) * sin(k_m)) * ((t / l) * t)) * (k_m / t)) * (k_m / t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 9e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)));
	elseif (l <= 1.02e+154)
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(l * l) / (sin(k_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(Float64(t / l) * t)) * Float64(k_m / t)) * Float64(k_m / t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 9e-161], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.02e+154], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 8.9999999999999993e-161

   1. Initial program 33.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.7%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 8.9999999999999993e-161 < l < 1.02000000000000007e154

   1. Initial program 34.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if 1.02000000000000007e154 < l

   1. Initial program 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 61.7

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.7

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

   6. Applied rewrites 76.7%

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

Alternative 8: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{t\_1}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{t\_1} \cdot \left(\left({\sin k\_m}^{2} \cdot \frac{k\_m}{\ell}\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 1.2e-60)
     (/
      2.0
      (/
       (*
        (*
         (*
          (fma
           (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
           (* k_m k_m)
           t)
          k_m)
         k_m)
        (/ k_m l))
       (/ t_1 k_m)))
     (/ 2.0 (* (/ k_m t_1) (* (* (pow (sin k_m) 2.0) (/ k_m l)) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 1.2e-60) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / (t_1 / k_m));
	} else {
		tmp = 2.0 / ((k_m / t_1) * ((pow(sin(k_m), 2.0) * (k_m / l)) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 1.2e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(t_1 / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / t_1) * Float64(Float64((sin(k_m) ^ 2.0) * Float64(k_m / l)) * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.2e-60], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / t$95$1), $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000005e-60

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.20000000000000005e-60 < k

   1. Initial program 28.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.6%

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

   11. Applied rewrites 97.6%

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

Alternative 9: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.05 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot {\sin k\_m}^{2}} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.05e-59)
   (/
    2.0
    (/
     (*
      (*
       (*
        (fma
         (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
         (* k_m k_m)
         t)
        k_m)
       k_m)
      (/ k_m l))
     (/ (* (cos k_m) l) k_m)))
   (*
    (/ (* 2.0 (cos k_m)) (* (* k_m k_m) (pow (sin k_m) 2.0)))
    (* l (/ l t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.05e-59) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
	} else {
		tmp = ((2.0 * cos(k_m)) / ((k_m * k_m) * pow(sin(k_m), 2.0))) * (l * (l / t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.05e-59)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k_m)) / Float64(Float64(k_m * k_m) * (sin(k_m) ^ 2.0))) * Float64(l * Float64(l / t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.05e-59], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.04999999999999998e-59

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.04999999999999998e-59 < k

   1. Initial program 28.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in t around 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/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. unpow2 N/A

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

      17. associate-/l* N/A

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

   10. Applied rewrites 77.1%

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

Alternative 10: 84.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 14600000000:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}}\\ \mathbf{elif}\;k\_m \leq 4.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\left(\tan k\_m \cdot \sin k\_m\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right) \cdot \frac{k\_m}{t}\right) \cdot \frac{k\_m}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \frac{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 14600000000.0)
   (/
    2.0
    (/
     (*
      (*
       (*
        (fma
         (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
         (* k_m k_m)
         t)
        k_m)
       k_m)
      (/ k_m l))
     (/ (* (cos k_m) l) k_m)))
   (if (<= k_m 4.8e+210)
     (/
      2.0
      (*
       (/ t l)
       (* (* (* (* (tan k_m) (sin k_m)) (* (/ t l) t)) (/ k_m t)) (/ k_m t))))
     (/ 2.0 (* (/ k_m l) (/ (* (* (pow (sin k_m) 2.0) t) k_m) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 14600000000.0) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
	} else if (k_m <= 4.8e+210) {
		tmp = 2.0 / ((t / l) * ((((tan(k_m) * sin(k_m)) * ((t / l) * t)) * (k_m / t)) * (k_m / t)));
	} else {
		tmp = 2.0 / ((k_m / l) * (((pow(sin(k_m), 2.0) * t) * k_m) / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 14600000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)));
	elseif (k_m <= 4.8e+210)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(Float64(Float64(tan(k_m) * sin(k_m)) * Float64(Float64(t / l) * t)) * Float64(k_m / t)) * Float64(k_m / t))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m) / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 14600000000.0], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.8e+210], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.46e10

   1. Initial program 38.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.8%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.7%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.46e10 < k < 4.79999999999999977e210

   1. Initial program 20.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 73.6

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lift-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 77.1

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

   6. Applied rewrites 77.1%

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

   if 4.79999999999999977e210 < k

   1. Initial program 37.6%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 92.3

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 69.1%

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

Alternative 11: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left({\sin k\_m}^{2} \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e-60)
   (/
    2.0
    (/
     (*
      (*
       (*
        (fma
         (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
         (* k_m k_m)
         t)
        k_m)
       k_m)
      (/ k_m l))
     (/ (* (cos k_m) l) k_m)))
   (/ 2.0 (* (/ k_m l) (* (pow (sin k_m) 2.0) (* t (/ k_m l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e-60) {
		tmp = 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
	} else {
		tmp = 2.0 / ((k_m / l) * (pow(sin(k_m), 2.0) * (t * (k_m / l))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64((sin(k_m) ^ 2.0) * Float64(t * Float64(k_m / l)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e-60], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-60

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.7%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
   9. Step-by-step derivation

   10. Applied rewrites 84.8%

       \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]

   if 1.25e-60 < k

   1. Initial program 28.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-sin.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 63.3%

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

Alternative 12: 74.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}}{\frac{\cos k\_m \cdot \ell}{k\_m}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (/
   (*
    (*
     (*
      (fma
       (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
       (* k_m k_m)
       t)
      k_m)
     k_m)
    (/ k_m l))
   (/ (* (cos k_m) l) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * (k_m / l)) / ((cos(k_m) * l) / k_m));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(k_m / l)) / Float64(Float64(cos(k_m) * l) / k_m)))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-sin.f64 92.5

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

5. Applied rewrites 92.5%

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

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
7. Step-by-step derivation

8. Applied rewrites 74.2%

   \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
9. Step-by-step derivation

10. Applied rewrites 76.4%

    \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \frac{k}{\ell}}{\color{blue}{\frac{\cos k \cdot \ell}{k}}}} \]
11. Add Preprocessing

Alternative 13: 74.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{\frac{k\_m}{\cos k\_m}}{\ell} \cdot \left(\frac{k\_m}{\ell} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (*
   (/ (/ k_m (cos k_m)) l)
   (*
    (/ k_m l)
    (*
     (*
      (fma
       (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
       (* k_m k_m)
       t)
      k_m)
     k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (((k_m / cos(k_m)) / l) * ((k_m / l) * ((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m)));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(k_m / cos(k_m)) / l) * Float64(Float64(k_m / l) * Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m))))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\color{blue}{\left({\sin k}^{2} \cdot t\right)} \cdot k}{\ell}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\color{blue}{\left({\sin k}^{2} \cdot t\right)} \cdot k}{\ell}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\color{blue}{{\sin k}^{2}} \cdot t\right) \cdot k}{\ell}} \]

   18. lower-sin.f64 92.5

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({\color{blue}{\sin k}}^{2} \cdot t\right) \cdot k}{\ell}} \]

5. Applied rewrites 92.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell}}} \]

6. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
7. Step-by-step derivation

8. Applied rewrites 74.2%

   \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]
9. Step-by-step derivation

10. Applied rewrites 76.3%

    \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right)}\right)} \]
11. Add Preprocessing

Alternative 14: 75.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k\_m \cdot k\_m, -0.3333333333333333\right) \cdot t, k\_m \cdot k\_m, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\left(k\_m \cdot \frac{k\_m}{\ell}\right) \cdot t}{2 \cdot \frac{\ell}{k\_m \cdot k\_m}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-164)
   (/
    2.0
    (*
     (*
      (*
       (fma
        (* (fma 0.044444444444444446 (* k_m k_m) -0.3333333333333333) t)
        (* k_m k_m)
        t)
       k_m)
      k_m)
     (* (/ k_m l) (/ k_m l))))
   (pow (/ (* (* k_m (/ k_m l)) t) (* 2.0 (/ l (* k_m k_m)))) -1.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-164) {
		tmp = 2.0 / (((fma((fma(0.044444444444444446, (k_m * k_m), -0.3333333333333333) * t), (k_m * k_m), t) * k_m) * k_m) * ((k_m / l) * (k_m / l)));
	} else {
		tmp = pow((((k_m * (k_m / l)) * t) / (2.0 * (l / (k_m * k_m)))), -1.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-164)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(0.044444444444444446, Float64(k_m * k_m), -0.3333333333333333) * t), Float64(k_m * k_m), t) * k_m) * k_m) * Float64(Float64(k_m / l) * Float64(k_m / l))));
	else
		tmp = Float64(Float64(Float64(k_m * Float64(k_m / l)) * t) / Float64(2.0 * Float64(l / Float64(k_m * k_m)))) ^ -1.0;
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-164], N[(2.0 / N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(k$95$m * k$95$m), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(2.0 * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2e-164

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}}{{\ell}^{2} \cdot \cos k}} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      6. times-frac N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\cos k \cdot \ell} \cdot \frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\cos k \cdot \ell} \cdot \frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\cos k}}{\ell}} \cdot \frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\cos k}}{\ell}} \cdot \frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{k}{\cos k}}}{\ell} \cdot \frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\color{blue}{\cos k}}}{\ell} \cdot \frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}} \]

      12. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot k}}{\ell}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \color{blue}{\frac{\left(t \cdot {\sin k}^{2}\right) \cdot k}{\ell}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot k}}{\ell}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\color{blue}{\left({\sin k}^{2} \cdot t\right)} \cdot k}{\ell}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\color{blue}{\left({\sin k}^{2} \cdot t\right)} \cdot k}{\ell}} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\color{blue}{{\sin k}^{2}} \cdot t\right) \cdot k}{\ell}} \]

      18. lower-sin.f64 94.6

          \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({\color{blue}{\sin k}}^{2} \cdot t\right) \cdot k}{\ell}} \]

   5. Applied rewrites 94.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left({k}^{2} \cdot \left(t + {k}^{2} \cdot \left(\frac{-1}{3} \cdot t + \frac{2}{45} \cdot \left({k}^{2} \cdot t\right)\right)\right)\right) \cdot k}{\ell}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.1%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \frac{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}{\ell}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k}{\ell} \cdot \frac{\color{blue}{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(\frac{2}{45}, k \cdot k, \frac{-1}{3}\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}}{\ell}} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.9%

       \[\leadsto \frac{2}{\frac{k}{\ell} \cdot \frac{\color{blue}{\left(\mathsf{fma}\left(t \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, t\right) \cdot \left(k \cdot k\right)\right) \cdot k}}{\ell}} \]
   12. Step-by-step derivation

   13. Applied rewrites 84.0%

       \[\leadsto \frac{2}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

   if 3.2e-164 < k

   1. Initial program 35.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

      10. lower-pow.f64 64.0

          \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.1%

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\frac{\ell}{k \cdot k}}{\color{blue}{k \cdot k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot t}{1 \cdot \left(2 \cdot \frac{\ell}{k \cdot k}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot t, k \cdot k, t\right) \cdot k\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot t}{2 \cdot \frac{\ell}{k \cdot k}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ {\left(\frac{\left(k\_m \cdot \frac{k\_m}{\ell}\right) \cdot t}{2 \cdot \frac{\ell}{k\_m \cdot k\_m}}\right)}^{-1} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (pow (/ (* (* k_m (/ k_m l)) t) (* 2.0 (/ l (* k_m k_m)))) -1.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return pow((((k_m * (k_m / l)) * t) / (2.0 * (l / (k_m * k_m)))), -1.0);
}

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 = (((k_m * (k_m / l)) * t) / (2.0d0 * (l / (k_m * k_m)))) ** (-1.0d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return Math.pow((((k_m * (k_m / l)) * t) / (2.0 * (l / (k_m * k_m)))), -1.0);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return math.pow((((k_m * (k_m / l)) * t) / (2.0 * (l / (k_m * k_m)))), -1.0)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(k_m * Float64(k_m / l)) * t) / Float64(2.0 * Float64(l / Float64(k_m * k_m)))) ^ -1.0
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (((k_m * (k_m / l)) * t) / (2.0 * (l / (k_m * k_m)))) ^ -1.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[Power[N[(N[(N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(2.0 * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

   10. lower-pow.f64 70.9

       \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
6. Step-by-step derivation

7. Applied rewrites 71.3%

   \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\frac{\ell}{k \cdot k}}{\color{blue}{k \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 75.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot t}{1 \cdot \left(2 \cdot \frac{\ell}{k \cdot k}\right)}}} \]

10. Final simplification 75.9%

    \[\leadsto {\left(\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot t}{2 \cdot \frac{\ell}{k \cdot k}}\right)}^{-1} \]
11. Add Preprocessing

Alternative 16: 72.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot 2\right) \cdot \frac{\frac{\ell}{k\_m}}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* (/ l (* (* k_m k_m) t)) 2.0) (/ (/ l k_m) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l / ((k_m * k_m) * t)) * 2.0) * ((l / k_m) / k_m);
}

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 / ((k_m * k_m) * t)) * 2.0d0) * ((l / k_m) / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l / ((k_m * k_m) * t)) * 2.0) * ((l / k_m) / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l / ((k_m * k_m) * t)) * 2.0) * ((l / k_m) / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l / Float64(Float64(k_m * k_m) * t)) * 2.0) * Float64(Float64(l / k_m) / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l / ((k_m * k_m) * t)) * 2.0) * ((l / k_m) / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

   10. lower-pow.f64 70.9

       \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
6. Step-by-step derivation

7. Applied rewrites 72.8%

   \[\leadsto \left(\frac{\ell \cdot 2}{t} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

8. Taylor expanded in t around 0

   \[\leadsto \left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 74.8%

    \[\leadsto \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot 2\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
11. Step-by-step derivation

12. Applied rewrites 74.9%

    \[\leadsto \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot 2\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]
13. Add Preprocessing

Alternative 17: 72.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\frac{\ell}{\left(t \cdot k\_m\right) \cdot k\_m} \cdot 2\right) \cdot \frac{\ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* (/ l (* (* t k_m) k_m)) 2.0) (/ l (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l / ((t * k_m) * k_m)) * 2.0) * (l / (k_m * k_m));
}

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 / ((t * k_m) * k_m)) * 2.0d0) * (l / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l / ((t * k_m) * k_m)) * 2.0) * (l / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l / ((t * k_m) * k_m)) * 2.0) * (l / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l / Float64(Float64(t * k_m) * k_m)) * 2.0) * Float64(l / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l / ((t * k_m) * k_m)) * 2.0) * (l / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l / N[(N[(t * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

   10. lower-pow.f64 70.9

       \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
6. Step-by-step derivation

7. Applied rewrites 72.8%

   \[\leadsto \left(\frac{\ell \cdot 2}{t} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

8. Taylor expanded in t around 0

   \[\leadsto \left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 74.8%

    \[\leadsto \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot 2\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
11. Step-by-step derivation

12. Applied rewrites 74.9%

    \[\leadsto \left(\frac{\ell}{\left(t \cdot k\right) \cdot k} \cdot 2\right) \cdot \frac{\ell}{k \cdot k} \]
13. Add Preprocessing

Alternative 18: 72.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot 2\right) \cdot \frac{\ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* (/ l (* (* k_m k_m) t)) 2.0) (/ l (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l / ((k_m * k_m) * t)) * 2.0) * (l / (k_m * k_m));
}

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 / ((k_m * k_m) * t)) * 2.0d0) * (l / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l / ((k_m * k_m) * t)) * 2.0) * (l / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l / ((k_m * k_m) * t)) * 2.0) * (l / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l / Float64(Float64(k_m * k_m) * t)) * 2.0) * Float64(l / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l / ((k_m * k_m) * t)) * 2.0) * (l / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

   10. lower-pow.f64 70.9

       \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
6. Step-by-step derivation

7. Applied rewrites 72.8%

   \[\leadsto \left(\frac{\ell \cdot 2}{t} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

8. Taylor expanded in t around 0

   \[\leadsto \left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 74.8%

    \[\leadsto \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot 2\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
11. Add Preprocessing

Alternative 19: 72.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot t}\right) \cdot \frac{\ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* l (/ 2.0 (* (* k_m k_m) t))) (/ l (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * (2.0 / ((k_m * k_m) * t))) * (l / (k_m * k_m));
}

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 * (2.0d0 / ((k_m * k_m) * t))) * (l / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * (2.0 / ((k_m * k_m) * t))) * (l / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * (2.0 / ((k_m * k_m) * t))) * (l / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * Float64(2.0 / Float64(Float64(k_m * k_m) * t))) * Float64(l / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * (2.0 / ((k_m * k_m) * t))) * (l / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t}} \cdot \frac{\ell}{{k}^{4}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t} \cdot \frac{\ell}{{k}^{4}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{4}}} \]

   10. lower-pow.f64 70.9

       \[\leadsto \frac{2 \cdot \ell}{t} \cdot \frac{\ell}{\color{blue}{{k}^{4}}} \]

5. Applied rewrites 70.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t} \cdot \frac{\ell}{{k}^{4}}} \]
6. Step-by-step derivation

7. Applied rewrites 72.8%

   \[\leadsto \left(\frac{\ell \cdot 2}{t} \cdot {k}^{-2}\right) \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

8. Taylor expanded in t around 0

   \[\leadsto \left(2 \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
9. Step-by-step derivation

10. Applied rewrites 74.8%

    \[\leadsto \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot 2\right) \cdot \frac{\color{blue}{\ell}}{k \cdot k} \]
11. Step-by-step derivation

12. Applied rewrites 74.8%

    \[\leadsto \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot t}\right) \cdot \frac{\ell}{k \cdot k} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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))))