Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 93.9%

Time: 11.3s

Alternatives: 17

Speedup: 7.8×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k\_m \leq 3.4 \cdot 10^{+78}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \ell}{k\_m} \cdot \frac{\ell \cdot 2}{\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.4e-130)
   (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))
   (if (<= k_m 3.4e+78)
     (/
      2.0
      (*
       (/ t l)
       (* (* (tan k_m) (/ (sin k_m) l)) (fma (* t t) 2.0 (* k_m k_m)))))
     (*
      (/ (* (cos k_m) l) k_m)
      (/ (* l 2.0) (* (* (pow (sin k_m) 2.0) t) k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.4e-130) {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	} else if (k_m <= 3.4e+78) {
		tmp = 2.0 / ((t / l) * ((tan(k_m) * (sin(k_m) / l)) * fma((t * t), 2.0, (k_m * k_m))));
	} else {
		tmp = ((cos(k_m) * l) / k_m) * ((l * 2.0) / ((pow(sin(k_m), 2.0) * t) * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.4e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	elseif (k_m <= 3.4e+78)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(tan(k_m) * Float64(sin(k_m) / l)) * fma(Float64(t * t), 2.0, Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) / k_m) * Float64(Float64(l * 2.0) / Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.4e-130], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.4e+78], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.3999999999999997e-130

   1. Initial program 54.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 75.4%

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

   if 4.3999999999999997e-130 < k < 3.40000000000000007e78

   1. Initial program 58.6%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 94.2%

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

   9. Applied rewrites 96.0%

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

   if 3.40000000000000007e78 < k

   1. Initial program 60.3%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 71.7%

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

   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 77.0%

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

   12. Applied rewrites 94.5%

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

Alternative 2: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq \infty:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m \cdot 2}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{k\_m}{\ell}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      INFINITY)
   (/ 2.0 (* (* (* (/ (* k_m 2.0) l) (* t t)) (/ k_m l)) t))
   (/ 2.0 (* (* (* k_m k_m) 2.0) (* (/ t l) (* (/ t l) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= ((double) INFINITY)) {
		tmp = 2.0 / (((((k_m * 2.0) / l) * (t * t)) * (k_m / l)) * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / (((((k_m * 2.0) / l) * (t * t)) * (k_m / l)) * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= math.inf:
		tmp = 2.0 / (((((k_m * 2.0) / l) * (t * t)) * (k_m / l)) * t)
	else:
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= Inf)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * 2.0) / l) * Float64(t * t)) * Float64(k_m / l)) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * 2.0) * Float64(Float64(t / l) * Float64(Float64(t / l) * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= Inf)
		tmp = 2.0 / (((((k_m * 2.0) / l) * (t * t)) * (k_m / l)) * t);
	else
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 84.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 74.7%

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

   9. Applied rewrites 84.2%

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

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 23.1

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

   5. Applied rewrites 23.1%

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

   7. Applied rewrites 19.7%

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

   9. Applied rewrites 40.8%

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

Alternative 3: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right) \leq \infty:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\frac{t}{\ell} \cdot \left(\frac{k\_m \cdot 2}{\ell} \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot 2\right) \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
       (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0))
      INFINITY)
   (/ 2.0 (* k_m (* (/ t l) (* (/ (* k_m 2.0) l) (* t t)))))
   (/ 2.0 (* (* (* k_m k_m) 2.0) (* (/ t l) (* (/ t l) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0)) <= ((double) INFINITY)) {
		tmp = 2.0 / (k_m * ((t / l) * (((k_m * 2.0) / l) * (t * t))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / (k_m * ((t / l) * (((k_m * 2.0) / l) * (t * t))));
	} else {
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0)) <= math.inf:
		tmp = 2.0 / (k_m * ((t / l) * (((k_m * 2.0) / l) * (t * t))))
	else:
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0)) <= Inf)
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(t / l) * Float64(Float64(Float64(k_m * 2.0) / l) * Float64(t * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * 2.0) * Float64(Float64(t / l) * Float64(Float64(t / l) * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0)) <= Inf)
		tmp = 2.0 / (k_m * ((t / l) * (((k_m * 2.0) / l) * (t * t))));
	else
		tmp = 2.0 / (((k_m * k_m) * 2.0) * ((t / l) * ((t / l) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 84.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 74.7%

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

   9. Applied rewrites 82.1%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 23.1

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

   5. Applied rewrites 23.1%

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

   7. Applied rewrites 19.7%

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

   9. Applied rewrites 40.8%

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

Alternative 4: 87.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k\_m \leq 9.6 \cdot 10^{+126}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\ell \cdot \ell\right) \cdot \cos k\_m\right) \cdot 2}{\left(\left({\sin k\_m}^{2} \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.4e-130)
   (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))
   (if (<= k_m 9.6e+126)
     (/
      2.0
      (*
       (/ t l)
       (* (* (tan k_m) (/ (sin k_m) l)) (fma (* t t) 2.0 (* k_m k_m)))))
     (/
      (* (* (* l l) (cos k_m)) 2.0)
      (* (* (* (pow (sin k_m) 2.0) t) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.4e-130) {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	} else if (k_m <= 9.6e+126) {
		tmp = 2.0 / ((t / l) * ((tan(k_m) * (sin(k_m) / l)) * fma((t * t), 2.0, (k_m * k_m))));
	} else {
		tmp = (((l * l) * cos(k_m)) * 2.0) / (((pow(sin(k_m), 2.0) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.4e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	elseif (k_m <= 9.6e+126)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(tan(k_m) * Float64(sin(k_m) / l)) * fma(Float64(t * t), 2.0, Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) * cos(k_m)) * 2.0) / Float64(Float64(Float64((sin(k_m) ^ 2.0) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.4e-130], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 9.6e+126], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.3999999999999997e-130

   1. Initial program 54.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 75.4%

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

   if 4.3999999999999997e-130 < k < 9.60000000000000048e126

   1. Initial program 60.4%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 92.5%

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

   9. Applied rewrites 95.3%

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

   if 9.60000000000000048e126 < k

   1. Initial program 58.2%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 68.1%

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

   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 77.2%

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

   12. Applied rewrites 77.2%

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

Alternative 5: 85.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\\ \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k\_m \leq 0.125:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{k\_m \cdot k\_m}{\ell}, \frac{0.08611111111111111}{\ell}\right) \cdot k\_m, k\_m, \frac{0.16666666666666666}{\ell}\right), k\_m \cdot k\_m, {\ell}^{-1}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k\_m \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{\left(\tan k\_m \cdot \sin k\_m\right) \cdot t}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (fma (* t t) 2.0 (* k_m k_m))))
   (if (<= k_m 4.4e-130)
     (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))
     (if (<= k_m 0.125)
       (/
        2.0
        (*
         t_1
         (*
          (*
           (fma
            (fma
             (*
              (fma
               0.03432539682539683
               (/ (* k_m k_m) l)
               (/ 0.08611111111111111 l))
              k_m)
             k_m
             (/ 0.16666666666666666 l))
            (* k_m k_m)
            (pow l -1.0))
           (* k_m k_m))
          (/ t l))))
       (if (<= k_m 3.8e+147)
         (/ 2.0 (* t_1 (/ (* (* (tan k_m) (sin k_m)) t) (* l l))))
         (/
          (* (* (* (cos k_m) l) l) 2.0)
          (* (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t) k_m) k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = fma((t * t), 2.0, (k_m * k_m));
	double tmp;
	if (k_m <= 4.4e-130) {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	} else if (k_m <= 0.125) {
		tmp = 2.0 / (t_1 * ((fma(fma((fma(0.03432539682539683, ((k_m * k_m) / l), (0.08611111111111111 / l)) * k_m), k_m, (0.16666666666666666 / l)), (k_m * k_m), pow(l, -1.0)) * (k_m * k_m)) * (t / l)));
	} else if (k_m <= 3.8e+147) {
		tmp = 2.0 / (t_1 * (((tan(k_m) * sin(k_m)) * t) / (l * l)));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (0.5 * cos((k_m + k_m)))) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = fma(Float64(t * t), 2.0, Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 4.4e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	elseif (k_m <= 0.125)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(fma(fma(Float64(fma(0.03432539682539683, Float64(Float64(k_m * k_m) / l), Float64(0.08611111111111111 / l)) * k_m), k_m, Float64(0.16666666666666666 / l)), Float64(k_m * k_m), (l ^ -1.0)) * Float64(k_m * k_m)) * Float64(t / l))));
	elseif (k_m <= 3.8e+147)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(Float64(tan(k_m) * sin(k_m)) * t) / Float64(l * l))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 4.4e-130], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.125], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(N[(N[(0.03432539682539683 * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] + N[(0.08611111111111111 / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + N[(0.16666666666666666 / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.8e+147], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 4.3999999999999997e-130

   1. Initial program 54.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 75.4%

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

   if 4.3999999999999997e-130 < k < 0.125

   1. Initial program 58.3%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{\ell} + \frac{31}{360} \cdot \frac{1}{\ell}\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right) + \frac{1}{\ell}\right)\right) \cdot \frac{\color{blue}{t}}{\ell}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.7%

       \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{k \cdot k}{\ell}, \frac{0.08611111111111111}{\ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell}\right), k \cdot k, \frac{1}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{t}}{\ell}\right)} \]

   if 0.125 < k < 3.7999999999999997e147

   1. Initial program 62.9%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 88.8%

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

   9. Applied rewrites 83.9%

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

   if 3.7999999999999997e147 < k

   1. Initial program 56.8%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 63.0%

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

   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 73.5%

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

   12. Applied rewrites 73.4%

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 0.125:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{k \cdot k}{\ell}, \frac{0.08611111111111111}{\ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell}\right), k \cdot k, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \frac{\left(\tan k \cdot \sin k\right) \cdot t}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k\_m \leq 9.6 \cdot 10^{+126}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\tan k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.4e-130)
   (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))
   (if (<= k_m 9.6e+126)
     (/
      2.0
      (*
       (/ t l)
       (* (* (tan k_m) (/ (sin k_m) l)) (fma (* t t) 2.0 (* k_m k_m)))))
     (/
      (* (* (* (cos k_m) l) l) 2.0)
      (* (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.4e-130) {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	} else if (k_m <= 9.6e+126) {
		tmp = 2.0 / ((t / l) * ((tan(k_m) * (sin(k_m) / l)) * fma((t * t), 2.0, (k_m * k_m))));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (0.5 * cos((k_m + k_m)))) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.4e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	elseif (k_m <= 9.6e+126)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(tan(k_m) * Float64(sin(k_m) / l)) * fma(Float64(t * t), 2.0, Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.4e-130], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 9.6e+126], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.3999999999999997e-130

   1. Initial program 54.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 75.4%

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

   if 4.3999999999999997e-130 < k < 9.60000000000000048e126

   1. Initial program 60.4%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 92.5%

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

   9. Applied rewrites 95.3%

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

   if 9.60000000000000048e126 < k

   1. Initial program 58.2%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 68.1%

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

   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 77.2%

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

   12. Applied rewrites 77.1%

       \[\leadsto \frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k\_m \leq 15000000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{k\_m \cdot k\_m}{\ell}, \frac{0.08611111111111111}{\ell}\right) \cdot k\_m, k\_m, \frac{0.16666666666666666}{\ell}\right), k\_m \cdot k\_m, {\ell}^{-1}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.4e-130)
   (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))
   (if (<= k_m 15000000000.0)
     (/
      2.0
      (*
       (fma (* t t) 2.0 (* k_m k_m))
       (*
        (*
         (fma
          (fma
           (*
            (fma
             0.03432539682539683
             (/ (* k_m k_m) l)
             (/ 0.08611111111111111 l))
            k_m)
           k_m
           (/ 0.16666666666666666 l))
          (* k_m k_m)
          (pow l -1.0))
         (* k_m k_m))
        (/ t l))))
     (/
      (* (* (* (cos k_m) l) l) 2.0)
      (* (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.4e-130) {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	} else if (k_m <= 15000000000.0) {
		tmp = 2.0 / (fma((t * t), 2.0, (k_m * k_m)) * ((fma(fma((fma(0.03432539682539683, ((k_m * k_m) / l), (0.08611111111111111 / l)) * k_m), k_m, (0.16666666666666666 / l)), (k_m * k_m), pow(l, -1.0)) * (k_m * k_m)) * (t / l)));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (0.5 * cos((k_m + k_m)))) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.4e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	elseif (k_m <= 15000000000.0)
		tmp = Float64(2.0 / Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(Float64(fma(fma(Float64(fma(0.03432539682539683, Float64(Float64(k_m * k_m) / l), Float64(0.08611111111111111 / l)) * k_m), k_m, Float64(0.16666666666666666 / l)), Float64(k_m * k_m), (l ^ -1.0)) * Float64(k_m * k_m)) * Float64(t / l))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.4e-130], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 15000000000.0], N[(2.0 / N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(0.03432539682539683 * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] + N[(0.08611111111111111 / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + N[(0.16666666666666666 / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.3999999999999997e-130

   1. Initial program 54.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 75.4%

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

   if 4.3999999999999997e-130 < k < 1.5e10

   1. Initial program 59.1%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left({k}^{2} \cdot \left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{173}{5040} \cdot \frac{{k}^{2}}{\ell} + \frac{31}{360} \cdot \frac{1}{\ell}\right) + \frac{1}{6} \cdot \frac{1}{\ell}\right) + \frac{1}{\ell}\right)\right) \cdot \frac{\color{blue}{t}}{\ell}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.1%

       \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{k \cdot k}{\ell}, \frac{0.08611111111111111}{\ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell}\right), k \cdot k, \frac{1}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{t}}{\ell}\right)} \]

   if 1.5e10 < k

   1. Initial program 59.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 75.4%

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

   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 75.4%

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

   12. Applied rewrites 75.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 15000000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.03432539682539683, \frac{k \cdot k}{\ell}, \frac{0.08611111111111111}{\ell}\right) \cdot k, k, \frac{0.16666666666666666}{\ell}\right), k \cdot k, {\ell}^{-1}\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t\right) \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\left(\left(\mathsf{fma}\left(\frac{k\_m \cdot k\_m}{\ell}, 0.16666666666666666, {\ell}^{-1}\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.2e-54)
   (/
    2.0
    (*
     (fma (* t t) 2.0 (* k_m k_m))
     (*
      (* (* (fma (/ (* k_m k_m) l) 0.16666666666666666 (pow l -1.0)) k_m) k_m)
      (/ t l))))
   (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.2e-54) {
		tmp = 2.0 / (fma((t * t), 2.0, (k_m * k_m)) * (((fma(((k_m * k_m) / l), 0.16666666666666666, pow(l, -1.0)) * k_m) * k_m) * (t / l)));
	} else {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.2e-54)
		tmp = Float64(2.0 / Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) / l), 0.16666666666666666, (l ^ -1.0)) * k_m) * k_m) * Float64(t / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.2e-54], N[(2.0 / N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2e-54

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right) \cdot \frac{\color{blue}{t}}{\ell}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.4%

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

   if 2.2e-54 < t

   1. Initial program 73.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 85.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left(\left(\mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, {\ell}^{-1}\right) \cdot k\right) \cdot k\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \frac{\frac{\left(k\_m \cdot t\right) \cdot k\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot 2\right) \cdot t}{\frac{\ell}{t}} \cdot \frac{k\_m}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.8e-63)
   (/ 2.0 (* (fma (* t t) 2.0 (* k_m k_m)) (/ (/ (* (* k_m t) k_m) l) l)))
   (/ 2.0 (* (/ (* (* k_m 2.0) t) (/ l t)) (/ k_m (/ l t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.8e-63) {
		tmp = 2.0 / (fma((t * t), 2.0, (k_m * k_m)) * ((((k_m * t) * k_m) / l) / l));
	} else {
		tmp = 2.0 / ((((k_m * 2.0) * t) / (l / t)) * (k_m / (l / t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.8e-63)
		tmp = Float64(2.0 / Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(Float64(Float64(Float64(k_m * t) * k_m) / l) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * 2.0) * t) / Float64(l / t)) * Float64(k_m / Float64(l / t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.8e-63], N[(2.0 / N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] * t), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999995e-63

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 82.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.1%

       \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \frac{\frac{\left(k \cdot t\right) \cdot k}{\ell}}{\ell}} \]

   if 5.7999999999999995e-63 < t

   1. Initial program 73.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 85.1%

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

Alternative 10: 71.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \frac{\frac{\left(k\_m \cdot t\right) \cdot k\_m}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{k\_m}{\frac{\ell}{t}}\right) \cdot \left(k\_m \cdot t\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.8e-63)
   (/ 2.0 (* (fma (* t t) 2.0 (* k_m k_m)) (/ (/ (* (* k_m t) k_m) l) l)))
   (/ 2.0 (* (* (* 2.0 (/ k_m (/ l t))) (* k_m t)) (/ t l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.8e-63) {
		tmp = 2.0 / (fma((t * t), 2.0, (k_m * k_m)) * ((((k_m * t) * k_m) / l) / l));
	} else {
		tmp = 2.0 / (((2.0 * (k_m / (l / t))) * (k_m * t)) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.8e-63)
		tmp = Float64(2.0 / Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(Float64(Float64(Float64(k_m * t) * k_m) / l) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * Float64(k_m / Float64(l / t))) * Float64(k_m * t)) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.8e-63], N[(2.0 / N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k$95$m * t), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999995e-63

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 82.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.1%

       \[\leadsto \frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \frac{\frac{\left(k \cdot t\right) \cdot k}{\ell}}{\ell}} \]

   if 5.7999999999999995e-63 < t

   1. Initial program 73.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 82.5%

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

Alternative 11: 70.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(t \cdot \frac{\frac{k\_m \cdot k\_m}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{k\_m}{\frac{\ell}{t}}\right) \cdot \left(k\_m \cdot t\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.2e-7)
   (/ 2.0 (* (fma (* t t) 2.0 (* k_m k_m)) (* t (/ (/ (* k_m k_m) l) l))))
   (/ 2.0 (* (* (* 2.0 (/ k_m (/ l t))) (* k_m t)) (/ t l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.2e-7) {
		tmp = 2.0 / (fma((t * t), 2.0, (k_m * k_m)) * (t * (((k_m * k_m) / l) / l)));
	} else {
		tmp = 2.0 / (((2.0 * (k_m / (l / t))) * (k_m * t)) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.2e-7)
		tmp = Float64(2.0 / Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(t * Float64(Float64(Float64(k_m * k_m) / l) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * Float64(k_m / Float64(l / t))) * Float64(k_m * t)) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.2e-7], N[(2.0 / N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.2e-7

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 82.1%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 67.2%

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

   if 4.2e-7 < t

   1. Initial program 70.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 62.1%

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

   9. Applied rewrites 85.3%

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

Alternative 12: 68.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m \cdot k\_m}{\ell}}{\ell} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(2 \cdot \frac{k\_m}{\frac{\ell}{t}}\right) \cdot \left(k\_m \cdot t\right)\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.2e-7)
   (/ 2.0 (* (/ (/ (* k_m k_m) l) l) (* t (fma (* t t) 2.0 (* k_m k_m)))))
   (/ 2.0 (* (* (* 2.0 (/ k_m (/ l t))) (* k_m t)) (/ t l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.2e-7) {
		tmp = 2.0 / ((((k_m * k_m) / l) / l) * (t * fma((t * t), 2.0, (k_m * k_m))));
	} else {
		tmp = 2.0 / (((2.0 * (k_m / (l / t))) * (k_m * t)) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.2e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / l) / l) * Float64(t * fma(Float64(t * t), 2.0, Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * Float64(k_m / Float64(l / t))) * Float64(k_m * t)) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.2e-7], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t * N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.2e-7

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 66.3%

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

   if 4.2e-7 < t

   1. Initial program 70.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 62.1%

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

   9. Applied rewrites 85.3%

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

Alternative 13: 66.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m \cdot k\_m}{\ell}}{\ell} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{k\_m}{\frac{\ell}{t}}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 8.2e-61)
   (/ 2.0 (* (/ (/ (* k_m k_m) l) l) (* t (fma (* t t) 2.0 (* k_m k_m)))))
   (/ 2.0 (* (* k_m 2.0) (* (* (/ t l) t) (/ k_m (/ l t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8.2e-61) {
		tmp = 2.0 / ((((k_m * k_m) / l) / l) * (t * fma((t * t), 2.0, (k_m * k_m))));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (((t / l) * t) * (k_m / (l / t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 8.2e-61)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / l) / l) * Float64(t * fma(Float64(t * t), 2.0, Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(Float64(Float64(t / l) * t) * Float64(k_m / Float64(l / t)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 8.2e-61], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t * N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.19999999999999998e-61

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 66.7%

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

   if 8.19999999999999998e-61 < t

   1. Initial program 73.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 79.5%

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

Alternative 14: 66.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{\frac{\frac{k\_m \cdot k\_m}{\ell}}{\ell} \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m \cdot 2}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{k\_m}{\ell}\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.25e-66)
   (/ 2.0 (* (/ (/ (* k_m k_m) l) l) (* t (fma (* t t) 2.0 (* k_m k_m)))))
   (/ 2.0 (* (* (* (/ (* k_m 2.0) l) (* t t)) (/ k_m l)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.25e-66) {
		tmp = 2.0 / ((((k_m * k_m) / l) / l) * (t * fma((t * t), 2.0, (k_m * k_m))));
	} else {
		tmp = 2.0 / (((((k_m * 2.0) / l) * (t * t)) * (k_m / l)) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.25e-66)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) / l) / l) * Float64(t * fma(Float64(t * t), 2.0, Float64(k_m * k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * 2.0) / l) * Float64(t * t)) * Float64(k_m / l)) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.25e-66], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(t * N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2499999999999999e-66

   1. Initial program 50.7%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 66.7%

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

   if 2.2499999999999999e-66 < t

   1. Initial program 73.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 61.4%

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

   9. Applied rewrites 80.7%

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

Alternative 15: 68.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m \cdot 2}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{k\_m}{\ell}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\left(\frac{t}{\ell \cdot \ell} \cdot k\_m\right) \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.8e-130)
   (/ 2.0 (* (* (* (/ (* k_m 2.0) l) (* t t)) (/ k_m l)) t))
   (/ 2.0 (* (fma (* t t) 2.0 (* k_m k_m)) (* (* (/ t (* l l)) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.8e-130) {
		tmp = 2.0 / (((((k_m * 2.0) / l) * (t * t)) * (k_m / l)) * t);
	} else {
		tmp = 2.0 / (fma((t * t), 2.0, (k_m * k_m)) * (((t / (l * l)) * k_m) * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.8e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * 2.0) / l) * Float64(t * t)) * Float64(k_m / l)) * t));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(Float64(Float64(t / Float64(l * l)) * k_m) * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.8e-130], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * 2.0), $MachinePrecision] / l), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.79999999999999993e-130

   1. Initial program 54.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 51.3%

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

   9. Applied rewrites 69.4%

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

   if 4.79999999999999993e-130 < k

   1. Initial program 59.5%

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

   3. Taylor expanded in t around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 82.8%

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

   9. Applied rewrites 82.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 67.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k \cdot 2}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\ell}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 57.8

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

5. Applied rewrites 57.8%

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

7. Applied rewrites 54.6%

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

9. Applied rewrites 65.3%

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

Alternative 17: 53.5% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-pow.f64 57.8

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

5. Applied rewrites 57.8%

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

7. Applied rewrites 54.6%

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

9. Applied rewrites 57.8%

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

Reproduce

herbie shell --seed 2024302 
(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))))