Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 89.4%

Time: 15.7s

Alternatives: 21

Speedup: 12.5×

• 27.6% 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: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(\tan k\_m \cdot k\_m\right) \cdot \frac{k\_m}{\ell}, \sin k\_m, \left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot 2\right)\right) \cdot t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.02e-100)
   (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)
   (/
    2.0
    (/
     (*
      (fma
       (* (* (tan k_m) k_m) (/ k_m l))
       (sin k_m)
       (* (* (sin k_m) (tan k_m)) (* (/ (* t t) l) 2.0)))
      t)
     l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e-100) {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	} else {
		tmp = 2.0 / ((fma(((tan(k_m) * k_m) * (k_m / l)), sin(k_m), ((sin(k_m) * tan(k_m)) * (((t * t) / l) * 2.0))) * t) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.02e-100)
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(tan(k_m) * k_m) * Float64(k_m / l)), sin(k_m), Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64(Float64(t * t) / l) * 2.0))) * t) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.02e-100], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Tan[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision] + N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.02e-100

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 69.9%

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

   11. Applied rewrites 73.0%

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

   if 1.02e-100 < k

   1. Initial program 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 82.7%

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

   9. Applied rewrites 87.0%

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

   11. Applied rewrites 87.0%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\frac{k}{\ell} \cdot \left(\tan k \cdot k\right), \sin k, \left(2 \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot t}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(\tan k \cdot k\right) \cdot \frac{k}{\ell}, \sin k, \left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot 2\right)\right) \cdot t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \tan k\_m\\ \mathbf{if}\;k\_m \leq 6 \cdot 10^{-93}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k\_m}{\ell} \cdot k\_m, t\_1, t\_1 \cdot \left(\frac{t \cdot t}{\ell} \cdot 2\right)\right) \cdot t} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (tan k_m))))
   (if (<= k_m 6e-93)
     (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)
     (*
      (/ 2.0 (* (fma (* (/ k_m l) k_m) t_1 (* t_1 (* (/ (* t t) l) 2.0))) t))
      l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * tan(k_m);
	double tmp;
	if (k_m <= 6e-93) {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	} else {
		tmp = (2.0 / (fma(((k_m / l) * k_m), t_1, (t_1 * (((t * t) / l) * 2.0))) * t)) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * tan(k_m))
	tmp = 0.0
	if (k_m <= 6e-93)
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	else
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(Float64(k_m / l) * k_m), t_1, Float64(t_1 * Float64(Float64(Float64(t * t) / l) * 2.0))) * t)) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 6e-93], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t$95$1 + N[(t$95$1 * N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.0000000000000003e-93

   1. Initial program 52.7%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.1

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

   5. Applied rewrites 53.1%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 70.0%

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

   11. Applied rewrites 73.2%

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

   if 6.0000000000000003e-93 < k

   1. Initial program 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 86.9%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

   11. Applied rewrites 87.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-93}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{k}{\ell} \cdot k, \sin k \cdot \tan k, \left(\sin k \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot 2\right)\right) \cdot t} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ t_2 := t\_1 \cdot 2\\ \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \mathbf{elif}\;k\_m \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_1, \frac{1}{\ell}\right) \cdot k\_m, k\_m, t\_2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}{\ell}}\\ \mathbf{elif}\;k\_m \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k\_m \cdot \tan k\_m}{\ell \cdot \ell} \cdot k\_m\right) \cdot \left(\mathsf{fma}\left(\frac{t}{k\_m \cdot k\_m} \cdot 2, t \cdot t, t\right) \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(t\_2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)) (t_2 (* t_1 2.0)))
   (if (<= k_m 1.02e-100)
     (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)
     (if (<= k_m 1.12e-7)
       (/
        2.0
        (/
         (*
          (*
           (fma (* (fma 0.3333333333333333 t_1 (/ 1.0 l)) k_m) k_m t_2)
           (* k_m k_m))
          t)
         l))
       (if (<= k_m 2e+156)
         (/
          2.0
          (*
           (* (/ (* (sin k_m) (tan k_m)) (* l l)) k_m)
           (* (fma (* (/ t (* k_m k_m)) 2.0) (* t t) t) k_m)))
         (/
          2.0
          (/
           (*
            (*
             (fma
              t_2
              t
              (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
             k_m)
            k_m)
           l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double t_2 = t_1 * 2.0;
	double tmp;
	if (k_m <= 1.02e-100) {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	} else if (k_m <= 1.12e-7) {
		tmp = 2.0 / (((fma((fma(0.3333333333333333, t_1, (1.0 / l)) * k_m), k_m, t_2) * (k_m * k_m)) * t) / l);
	} else if (k_m <= 2e+156) {
		tmp = 2.0 / ((((sin(k_m) * tan(k_m)) / (l * l)) * k_m) * (fma(((t / (k_m * k_m)) * 2.0), (t * t), t) * k_m));
	} else {
		tmp = 2.0 / (((fma(t_2, t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	t_2 = Float64(t_1 * 2.0)
	tmp = 0.0
	if (k_m <= 1.02e-100)
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	elseif (k_m <= 1.12e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(0.3333333333333333, t_1, Float64(1.0 / l)) * k_m), k_m, t_2) * Float64(k_m * k_m)) * t) / l));
	elseif (k_m <= 2e+156)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k_m) * tan(k_m)) / Float64(l * l)) * k_m) * Float64(fma(Float64(Float64(t / Float64(k_m * k_m)) * 2.0), Float64(t * t), t) * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(t_2, t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	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] / l), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 2.0), $MachinePrecision]}, If[LessEqual[k$95$m, 1.02e-100], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[k$95$m, 1.12e-7], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * t$95$1 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + t$95$2), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2e+156], N[(2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(N[(N[(t / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$2 * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.02e-100

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 69.9%

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

   11. Applied rewrites 73.0%

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

   if 1.02e-100 < k < 1.12e-7

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 95.3%

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

   9. Applied rewrites 95.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 95.4%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \frac{t \cdot t}{\ell}, \frac{1}{\ell}\right) \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell}} \]

   if 1.12e-7 < k < 2e156

   1. Initial program 40.2%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 93.4%

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

   if 2e156 < k

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \frac{t \cdot t}{\ell}, \frac{1}{\ell}\right) \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \tan k}{\ell \cdot \ell} \cdot k\right) \cdot \left(\mathsf{fma}\left(\frac{t}{k \cdot k} \cdot 2, t \cdot t, t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ t_2 := t\_1 \cdot 2\\ \mathbf{if}\;k\_m \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \mathbf{elif}\;k\_m \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_1, \frac{1}{\ell}\right) \cdot k\_m, k\_m, t\_2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}{\ell}}\\ \mathbf{elif}\;k\_m \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k\_m}{\ell \cdot \ell} \cdot \tan k\_m\right) \cdot k\_m\right) \cdot \left(\mathsf{fma}\left(\frac{t}{k\_m \cdot k\_m} \cdot 2, t \cdot t, t\right) \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(t\_2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)) (t_2 (* t_1 2.0)))
   (if (<= k_m 1.02e-100)
     (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)
     (if (<= k_m 1.12e-7)
       (/
        2.0
        (/
         (*
          (*
           (fma (* (fma 0.3333333333333333 t_1 (/ 1.0 l)) k_m) k_m t_2)
           (* k_m k_m))
          t)
         l))
       (if (<= k_m 2e+156)
         (/
          2.0
          (*
           (* (* (/ (sin k_m) (* l l)) (tan k_m)) k_m)
           (* (fma (* (/ t (* k_m k_m)) 2.0) (* t t) t) k_m)))
         (/
          2.0
          (/
           (*
            (*
             (fma
              t_2
              t
              (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
             k_m)
            k_m)
           l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double t_2 = t_1 * 2.0;
	double tmp;
	if (k_m <= 1.02e-100) {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	} else if (k_m <= 1.12e-7) {
		tmp = 2.0 / (((fma((fma(0.3333333333333333, t_1, (1.0 / l)) * k_m), k_m, t_2) * (k_m * k_m)) * t) / l);
	} else if (k_m <= 2e+156) {
		tmp = 2.0 / ((((sin(k_m) / (l * l)) * tan(k_m)) * k_m) * (fma(((t / (k_m * k_m)) * 2.0), (t * t), t) * k_m));
	} else {
		tmp = 2.0 / (((fma(t_2, t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	t_2 = Float64(t_1 * 2.0)
	tmp = 0.0
	if (k_m <= 1.02e-100)
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	elseif (k_m <= 1.12e-7)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(fma(0.3333333333333333, t_1, Float64(1.0 / l)) * k_m), k_m, t_2) * Float64(k_m * k_m)) * t) / l));
	elseif (k_m <= 2e+156)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k_m) / Float64(l * l)) * tan(k_m)) * k_m) * Float64(fma(Float64(Float64(t / Float64(k_m * k_m)) * 2.0), Float64(t * t), t) * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(t_2, t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	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] / l), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 2.0), $MachinePrecision]}, If[LessEqual[k$95$m, 1.02e-100], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[k$95$m, 1.12e-7], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * t$95$1 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + t$95$2), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2e+156], N[(2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(N[(N[(t / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$2 * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.02e-100

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 69.9%

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

   11. Applied rewrites 73.0%

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

   if 1.02e-100 < k < 1.12e-7

   1. Initial program 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 95.3%

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

   9. Applied rewrites 95.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 95.4%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \frac{t \cdot t}{\ell}, \frac{1}{\ell}\right) \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell}} \]

   if 1.12e-7 < k < 2e156

   1. Initial program 40.2%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 90.1%

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

   9. Applied rewrites 93.2%

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

   if 2e156 < k

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \frac{t \cdot t}{\ell}, \frac{1}{\ell}\right) \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\ell}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot k\right) \cdot \left(\mathsf{fma}\left(\frac{t}{k \cdot k} \cdot 2, t \cdot t, t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \mathbf{elif}\;k\_m \leq 7.2 \cdot 10^{+240}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t}{k\_m \cdot k\_m} \cdot 2, t \cdot t, t\right) \cdot k\_m\right) \cdot \left(\frac{\sin k\_m}{\ell} \cdot \tan k\_m\right)}{\ell} \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k\_m \cdot k\_m}{\ell} \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e-73)
   (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)
   (if (<= k_m 7.2e+240)
     (/
      2.0
      (*
       (/
        (*
         (* (fma (* (/ t (* k_m k_m)) 2.0) (* t t) t) k_m)
         (* (/ (sin k_m) l) (tan k_m)))
        l)
       k_m))
     (/ 2.0 (/ (* (* (/ (* k_m k_m) l) t) (* k_m k_m)) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e-73) {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	} else if (k_m <= 7.2e+240) {
		tmp = 2.0 / ((((fma(((t / (k_m * k_m)) * 2.0), (t * t), t) * k_m) * ((sin(k_m) / l) * tan(k_m))) / l) * k_m);
	} else {
		tmp = 2.0 / (((((k_m * k_m) / l) * t) * (k_m * k_m)) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e-73)
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	elseif (k_m <= 7.2e+240)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(t / Float64(k_m * k_m)) * 2.0), Float64(t * t), t) * k_m) * Float64(Float64(sin(k_m) / l) * tan(k_m))) / l) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) / l) * t) * Float64(k_m * k_m)) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.2e-73], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[k$95$m, 7.2e+240], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.2e-73

   1. Initial program 53.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   7. Applied rewrites 66.2%

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

   9. Applied rewrites 70.6%

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

   11. Applied rewrites 73.6%

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

   if 2.2e-73 < k < 7.1999999999999997e240

   1. Initial program 50.3%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 89.7%

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

   if 7.1999999999999997e240 < k

   1. Initial program 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 64.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+240}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t}{k \cdot k} \cdot 2, t \cdot t, t\right) \cdot k\right) \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)}{\ell} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot \left(k \cdot k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \mathbf{elif}\;k\_m \leq 3.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(\frac{t}{k\_m \cdot k\_m} \cdot 2, t \cdot t, t\right) \cdot \left(\frac{\sin k\_m}{\ell} \cdot \tan k\_m\right)}{\ell} \cdot k\_m\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e-73)
   (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)
   (if (<= k_m 3.7e+155)
     (/
      2.0
      (*
       (*
        (/
         (*
          (fma (* (/ t (* k_m k_m)) 2.0) (* t t) t)
          (* (/ (sin k_m) l) (tan k_m)))
         l)
        k_m)
       k_m))
     (/
      2.0
      (/
       (*
        (*
         (fma
          (* (/ (* t t) l) 2.0)
          t
          (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
         k_m)
        k_m)
       l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e-73) {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	} else if (k_m <= 3.7e+155) {
		tmp = 2.0 / ((((fma(((t / (k_m * k_m)) * 2.0), (t * t), t) * ((sin(k_m) / l) * tan(k_m))) / l) * k_m) * k_m);
	} else {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e-73)
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	elseif (k_m <= 3.7e+155)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(Float64(t / Float64(k_m * k_m)) * 2.0), Float64(t * t), t) * Float64(Float64(sin(k_m) / l) * tan(k_m))) / l) * k_m) * k_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.2e-73], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[k$95$m, 3.7e+155], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + t), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.2e-73

   1. Initial program 53.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   7. Applied rewrites 66.2%

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

   9. Applied rewrites 70.6%

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

   11. Applied rewrites 73.6%

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

   if 2.2e-73 < k < 3.6999999999999998e155

   1. Initial program 46.9%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 91.0%

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

   if 3.6999999999999998e155 < k

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-73}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(\frac{t}{k \cdot k} \cdot 2, t \cdot t, t\right) \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)}{\ell} \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot t\\ t_2 := \mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right)\\ \mathbf{if}\;t \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k\_m}{\ell}, \left(\sin k\_m \cdot \tan k\_m\right) \cdot k\_m, \frac{\left(\left(t \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\ell} \cdot 2\right) \cdot t}{\ell}}\\ \mathbf{elif}\;t \leq 1.92 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_1 \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\_m\right) \cdot t\_2} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{\left(\left(t\_2 \cdot \tan k\_m\right) \cdot t\_1\right) \cdot \frac{t}{\ell}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) t)) (t_2 (fma (/ k_m (* t t)) k_m 2.0)))
   (if (<= t 2.5e-26)
     (/
      2.0
      (/
       (*
        (fma
         (/ k_m l)
         (* (* (sin k_m) (tan k_m)) k_m)
         (* (/ (* (* (* t t) k_m) k_m) l) 2.0))
        t)
       l))
     (if (<= t 1.92e+205)
       (* (/ 2.0 (* (* (* (* t_1 t) (/ t l)) (tan k_m)) t_2)) l)
       (/ (/ (* l 2.0) (* (* (* t_2 (tan k_m)) t_1) (/ t l))) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * t;
	double t_2 = fma((k_m / (t * t)), k_m, 2.0);
	double tmp;
	if (t <= 2.5e-26) {
		tmp = 2.0 / ((fma((k_m / l), ((sin(k_m) * tan(k_m)) * k_m), (((((t * t) * k_m) * k_m) / l) * 2.0)) * t) / l);
	} else if (t <= 1.92e+205) {
		tmp = (2.0 / ((((t_1 * t) * (t / l)) * tan(k_m)) * t_2)) * l;
	} else {
		tmp = ((l * 2.0) / (((t_2 * tan(k_m)) * t_1) * (t / l))) / t;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * t)
	t_2 = fma(Float64(k_m / Float64(t * t)), k_m, 2.0)
	tmp = 0.0
	if (t <= 2.5e-26)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(k_m / l), Float64(Float64(sin(k_m) * tan(k_m)) * k_m), Float64(Float64(Float64(Float64(Float64(t * t) * k_m) * k_m) / l) * 2.0)) * t) / l));
	elseif (t <= 1.92e+205)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(t_1 * t) * Float64(t / l)) * tan(k_m)) * t_2)) * l);
	else
		tmp = Float64(Float64(Float64(l * 2.0) / Float64(Float64(Float64(t_2 * tan(k_m)) * t_1) * Float64(t / l))) / t);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision]}, If[LessEqual[t, 2.5e-26], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] + N[(N[(N[(N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.92e+205], N[(N[(2.0 / N[(N[(N[(N[(t$95$1 * t), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[(N[(t$95$2 * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.5000000000000001e-26

   1. Initial program 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.8%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 81.0%

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

   9. Applied rewrites 83.1%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.0%

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

   if 2.5000000000000001e-26 < t < 1.92e205

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 82.6

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{\color{blue}{1.5}}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 82.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 76.9%

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

   if 1.92e205 < t

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 58.3

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{\color{blue}{1.5}}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 58.3%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 52.0%

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

   8. Applied rewrites 89.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{\ell}, \left(\sin k \cdot \tan k\right) \cdot k, \frac{\left(\left(t \cdot t\right) \cdot k\right) \cdot k}{\ell} \cdot 2\right) \cdot t}{\ell}}\\ \mathbf{elif}\;t \leq 1.92 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{\left(\left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \frac{t}{\ell}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+168}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\sin k\_m \cdot t\right) \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)}^{-1} \cdot \frac{1}{t}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6.4e-47)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (if (<= t 1.3e+168)
     (*
      (/
       2.0
       (*
        (* (* (* (* (sin k_m) t) t) (/ t l)) (tan k_m))
        (fma (/ k_m (* t t)) k_m 2.0)))
      l)
     (* (* (pow (* (* (/ t l) k_m) (* t k_m)) -1.0) (/ 1.0 t)) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6.4e-47) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else if (t <= 1.3e+168) {
		tmp = (2.0 / (((((sin(k_m) * t) * t) * (t / l)) * tan(k_m)) * fma((k_m / (t * t)), k_m, 2.0))) * l;
	} else {
		tmp = (pow((((t / l) * k_m) * (t * k_m)), -1.0) * (1.0 / t)) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6.4e-47)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	elseif (t <= 1.3e+168)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k_m) * t) * t) * Float64(t / l)) * tan(k_m)) * fma(Float64(k_m / Float64(t * t)), k_m, 2.0))) * l);
	else
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * Float64(t * k_m)) ^ -1.0) * Float64(1.0 / t)) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6.4e-47], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e+168], N[(N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.3999999999999998e-47

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 6.3999999999999998e-47 < t < 1.3e168

   1. Initial program 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 80.0

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{\color{blue}{1.5}}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 80.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 75.8%

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

   if 1.3e168 < t

   1. Initial program 55.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 61.3%

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

   9. Applied rewrites 65.0%

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

   11. Applied rewrites 75.6%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+168}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(t \cdot k\right)\right)}^{-1} \cdot \frac{1}{t}\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+147}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot t\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)}^{-1} \cdot \frac{1}{t}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-49)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (if (<= t 1e+147)
     (*
      (/
       2.0
       (*
        (* (* (fma (/ k_m (* t t)) k_m 2.0) (* (sin k_m) (tan k_m))) (/ t l))
        (* t t)))
      l)
     (* (* (pow (* (* (/ t l) k_m) (* t k_m)) -1.0) (/ 1.0 t)) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-49) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else if (t <= 1e+147) {
		tmp = (2.0 / (((fma((k_m / (t * t)), k_m, 2.0) * (sin(k_m) * tan(k_m))) * (t / l)) * (t * t))) * l;
	} else {
		tmp = (pow((((t / l) * k_m) * (t * k_m)), -1.0) * (1.0 / t)) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-49)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	elseif (t <= 1e+147)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(fma(Float64(k_m / Float64(t * t)), k_m, 2.0) * Float64(sin(k_m) * tan(k_m))) * Float64(t / l)) * Float64(t * t))) * l);
	else
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * Float64(t * k_m)) ^ -1.0) * Float64(1.0 / t)) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-49], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+147], N[(N[(2.0 / N[(N[(N[(N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999992e-49

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 2.39999999999999992e-49 < t < 9.9999999999999998e146

   1. Initial program 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 79.0

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{\color{blue}{1.5}}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 79.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 72.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.2

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

   8. Applied rewrites 72.2%

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

   if 9.9999999999999998e146 < t

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 60.7%

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

   9. Applied rewrites 64.1%

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

   11. Applied rewrites 74.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+147}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot t\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(t \cdot k\right)\right)}^{-1} \cdot \frac{1}{t}\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right) \cdot \tan k\_m\right) \cdot \left(\sin k\_m \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot t} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4e-44)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (*
    (/
     2.0
     (*
      (*
       (* (* (fma (/ k_m (* t t)) k_m 2.0) (tan k_m)) (* (sin k_m) t))
       (/ t l))
      t))
    l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4e-44) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else {
		tmp = (2.0 / ((((fma((k_m / (t * t)), k_m, 2.0) * tan(k_m)) * (sin(k_m) * t)) * (t / l)) * t)) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4e-44)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(k_m / Float64(t * t)), k_m, 2.0) * tan(k_m)) * Float64(sin(k_m) * t)) * Float64(t / l)) * t)) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4e-44], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999981e-44

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 3.99999999999999981e-44 < t

   1. Initial program 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 76.8

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{\color{blue}{1.5}}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 76.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 64.1%

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

   8. Applied rewrites 75.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \tan k\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \frac{t}{\ell}\right) \cdot t} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k\_m \cdot t\right) \cdot t\right) \cdot \left(\mathsf{fma}\left(\frac{k\_m}{t \cdot t}, k\_m, 2\right) \cdot \tan k\_m\right)\right) \cdot \frac{t}{\ell}} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4e-44)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (*
    (/
     2.0
     (*
      (* (* (* (sin k_m) t) t) (* (fma (/ k_m (* t t)) k_m 2.0) (tan k_m)))
      (/ t l)))
    l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4e-44) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else {
		tmp = (2.0 / ((((sin(k_m) * t) * t) * (fma((k_m / (t * t)), k_m, 2.0) * tan(k_m))) * (t / l))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4e-44)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(sin(k_m) * t) * t) * Float64(fma(Float64(k_m / Float64(t * t)), k_m, 2.0) * tan(k_m))) * Float64(t / l))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4e-44], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] * k$95$m + 2.0), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.99999999999999981e-44

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 3.99999999999999981e-44 < t

   1. Initial program 57.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. times-frac N/A

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

      9. times-frac N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 76.8

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{\color{blue}{1.5}}}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 76.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{\sin k}{\frac{\ell}{{t}^{1.5}}}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 73.0%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot t\right) \cdot t\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t \cdot t}, k, 2\right) \cdot \tan k\right)\right) \cdot \frac{t}{\ell}} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k\_m}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.8e-41)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (* (* (pow (* (* (/ t l) k_m) t) -1.0) (/ 1.0 (* t k_m))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.8e-41) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else {
		tmp = (pow((((t / l) * k_m) * t), -1.0) * (1.0 / (t * k_m))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.8e-41)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	else
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * t) ^ -1.0) * Float64(1.0 / Float64(t * k_m))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.8e-41], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.80000000000000044e-41

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 4.80000000000000044e-41 < t

   1. Initial program 57.2%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 66.0%

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

   11. Applied rewrites 71.6%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot t\right)}^{-1} \cdot \frac{1}{t \cdot k}\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)\right)}^{-1} \cdot \frac{1}{t}\right) \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.8e+63)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (* (* (pow (* (* (/ t l) k_m) (* t k_m)) -1.0) (/ 1.0 t)) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.8e+63) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else {
		tmp = (pow((((t / l) * k_m) * (t * k_m)), -1.0) * (1.0 / t)) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.8e+63)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	else
		tmp = Float64(Float64((Float64(Float64(Float64(t / l) * k_m) * Float64(t * k_m)) ^ -1.0) * Float64(1.0 / t)) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.8e+63], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(N[(t / l), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.79999999999999987e63

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.9%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 2.79999999999999987e63 < t

   1. Initial program 52.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 64.7%

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

   9. Applied rewrites 66.9%

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

   11. Applied rewrites 73.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(t \cdot k\right)\right)}^{-1} \cdot \frac{1}{t}\right) \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k\_m} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.8e-41)
   (/
    2.0
    (/
     (*
      (*
       (fma
        (* (/ (* t t) l) 2.0)
        t
        (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
       k_m)
      k_m)
     l))
   (if (<= t 3e+146)
     (* (/ l (* t k_m)) (/ l (* (* t t) k_m)))
     (* (/ l (* (* (* t k_m) t) (* t k_m))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.8e-41) {
		tmp = 2.0 / (((fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l);
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.8e-41)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * k_m) * k_m) / l));
	elseif (t <= 3e+146)
		tmp = Float64(Float64(l / Float64(t * k_m)) * Float64(l / Float64(Float64(t * t) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.8e-41], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e+146], N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.80000000000000044e-41

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot \color{blue}{k}}{\ell}} \]

   if 4.80000000000000044e-41 < t < 3.00000000000000002e146

   1. Initial program 60.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 69.2%

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

   9. Applied rewrites 74.3%

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

   if 3.00000000000000002e146 < t

   1. Initial program 53.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 62.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot k\right) \cdot k}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \ell\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k\_m} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.3e-27)
   (*
    (/
     2.0
     (*
      (fma
       (* (/ (* t t) l) 2.0)
       t
       (* (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) k_m) k_m))
      (* k_m k_m)))
    l)
   (if (<= t 3e+146)
     (* (/ l (* t k_m)) (/ l (* (* t t) k_m)))
     (* (/ l (* (* (* t k_m) t) (* t k_m))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.3e-27) {
		tmp = (2.0 / (fma((((t * t) / l) * 2.0), t, ((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * k_m) * k_m)) * (k_m * k_m))) * l;
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.3e-27)
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(Float64(Float64(t * t) / l) * 2.0), t, Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * k_m) * k_m)) * Float64(k_m * k_m))) * l);
	elseif (t <= 3e+146)
		tmp = Float64(Float64(l / Float64(t * k_m)) * Float64(l / Float64(Float64(t * t) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.3e-27], N[(N[(2.0 / N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t + N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], If[LessEqual[t, 3e+146], N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.2999999999999999e-27

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2 \cdot \frac{t \cdot t}{\ell}, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right)} \cdot \ell} \]

   if 2.2999999999999999e-27 < t < 3.00000000000000002e146

   1. Initial program 57.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 67.4%

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

   9. Applied rewrites 72.9%

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

   if 3.00000000000000002e146 < t

   1. Initial program 53.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 62.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{t \cdot t}{\ell} \cdot 2, t, \left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k\right) \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \ell\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \left(k\_m \cdot k\_m\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k\_m} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.2e-50)
   (/
    2.0
    (/
     (*
      (* (/ (* (fma 0.3333333333333333 (* t t) 1.0) t) l) (* k_m k_m))
      (* k_m k_m))
     l))
   (if (<= t 3e+146)
     (* (/ l (* t k_m)) (/ l (* (* t t) k_m)))
     (* (/ l (* (* (* t k_m) t) (* t k_m))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.2e-50) {
		tmp = 2.0 / (((((fma(0.3333333333333333, (t * t), 1.0) * t) / l) * (k_m * k_m)) * (k_m * k_m)) / l);
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.2e-50)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(0.3333333333333333, Float64(t * t), 1.0) * t) / l) * Float64(k_m * k_m)) * Float64(k_m * k_m)) / l));
	elseif (t <= 3e+146)
		tmp = Float64(Float64(l / Float64(t * k_m)) * Float64(l / Float64(Float64(t * t) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.2e-50], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e+146], N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.20000000000000001e-50

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 60.1%

       \[\leadsto \frac{2}{\frac{\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{k} \cdot k\right)}{\ell}} \]

   if 1.20000000000000001e-50 < t < 3.00000000000000002e146

   1. Initial program 60.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 69.2%

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

   9. Applied rewrites 74.3%

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

   if 3.00000000000000002e146 < t

   1. Initial program 53.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 58.8%

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

   9. Applied rewrites 62.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k\_m \cdot k\_m}{\ell} \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k\_m} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.2e-50)
   (/ 2.0 (/ (* (* (/ (* k_m k_m) l) t) (* k_m k_m)) l))
   (if (<= t 3e+146)
     (* (/ l (* t k_m)) (/ l (* (* t t) k_m)))
     (* (/ l (* (* (* t k_m) t) (* t k_m))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.2e-50) {
		tmp = 2.0 / (((((k_m * k_m) / l) * t) * (k_m * k_m)) / l);
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.2d-50) then
        tmp = 2.0d0 / (((((k_m * k_m) / l) * t) * (k_m * k_m)) / l)
    else if (t <= 3d+146) then
        tmp = (l / (t * k_m)) * (l / ((t * t) * k_m))
    else
        tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.2e-50) {
		tmp = 2.0 / (((((k_m * k_m) / l) * t) * (k_m * k_m)) / l);
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.2e-50:
		tmp = 2.0 / (((((k_m * k_m) / l) * t) * (k_m * k_m)) / l)
	elif t <= 3e+146:
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m))
	else:
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.2e-50)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) / l) * t) * Float64(k_m * k_m)) / l));
	elseif (t <= 3e+146)
		tmp = Float64(Float64(l / Float64(t * k_m)) * Float64(l / Float64(Float64(t * t) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.2e-50)
		tmp = 2.0 / (((((k_m * k_m) / l) * t) * (k_m * k_m)) / l);
	elseif (t <= 3e+146)
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	else
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.2e-50], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e+146], N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.20000000000000001e-50

   1. Initial program 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 64.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot t}{\ell} \cdot k, k, \frac{\left(t \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)}}{\ell}} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 57.4%

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

   if 1.20000000000000001e-50 < t < 3.00000000000000002e146

   1. Initial program 60.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 63.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.2%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.3%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

   if 3.00000000000000002e146 < t

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 50.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.8%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 68.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \ell}{t}}{t}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k\_m} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.2e-65)
   (/ (/ (* (/ l (* (* k_m k_m) t)) l) t) t)
   (if (<= t 3e+146)
     (* (/ l (* t k_m)) (/ l (* (* t t) k_m)))
     (* (/ l (* (* (* t k_m) t) (* t k_m))) l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.2e-65) {
		tmp = (((l / ((k_m * k_m) * t)) * l) / t) / t;
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.2d-65) then
        tmp = (((l / ((k_m * k_m) * t)) * l) / t) / t
    else if (t <= 3d+146) then
        tmp = (l / (t * k_m)) * (l / ((t * t) * k_m))
    else
        tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.2e-65) {
		tmp = (((l / ((k_m * k_m) * t)) * l) / t) / t;
	} else if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.2e-65:
		tmp = (((l / ((k_m * k_m) * t)) * l) / t) / t
	elif t <= 3e+146:
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m))
	else:
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.2e-65)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(k_m * k_m) * t)) * l) / t) / t);
	elseif (t <= 3e+146)
		tmp = Float64(Float64(l / Float64(t * k_m)) * Float64(l / Float64(Float64(t * t) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.2e-65)
		tmp = (((l / ((k_m * k_m) * t)) * l) / t) / t;
	elseif (t <= 3e+146)
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	else
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.2e-65], N[(N[(N[(N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 3e+146], N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.2000000000000001e-65

   1. Initial program 49.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 49.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.3%

      \[\leadsto \frac{\frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{t}}{\color{blue}{t}} \]

   if 1.2000000000000001e-65 < t < 3.00000000000000002e146

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 61.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.8%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.5%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

   if 3.00000000000000002e146 < t

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 50.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.8%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \ell}{t}}{t}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 69.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.4e-111)
   (* (/ l (* (* (* t k_m) t) (* t k_m))) l)
   (* (/ l t) (/ l (* (* (* k_m k_m) t) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e-111) {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	} else {
		tmp = (l / t) * (l / (((k_m * k_m) * t) * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.4d-111) then
        tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
    else
        tmp = (l / t) * (l / (((k_m * k_m) * t) * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.4e-111) {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	} else {
		tmp = (l / t) * (l / (((k_m * k_m) * t) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.4e-111:
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
	else:
		tmp = (l / t) * (l / (((k_m * k_m) * t) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.4e-111)
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(Float64(Float64(k_m * k_m) * t) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.4e-111)
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	else
		tmp = (l / t) * (l / (((k_m * k_m) * t) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.4e-111], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999997e-111

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 52.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.1%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

   if 3.39999999999999997e-111 < k

   1. Initial program 51.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 49.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.4%

      \[\leadsto \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 66.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k\_m} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3e+146)
   (* (/ l (* t k_m)) (/ l (* (* t t) k_m)))
   (* (/ l (* (* (* t k_m) t) (* t k_m))) l)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 3d+146) then
        tmp = (l / (t * k_m)) * (l / ((t * t) * k_m))
    else
        tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e+146) {
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	} else {
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3e+146:
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m))
	else:
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3e+146)
		tmp = Float64(Float64(l / Float64(t * k_m)) * Float64(l / Float64(Float64(t * t) * k_m)));
	else
		tmp = Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3e+146)
		tmp = (l / (t * k_m)) * (l / ((t * t) * k_m));
	else
		tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3e+146], N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.00000000000000002e146

   1. Initial program 51.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 51.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 63.6%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

   if 3.00000000000000002e146 < t

   1. Initial program 53.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

      12. lower-*.f64 50.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.8%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.0%

      \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{+146}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot \left(t \cdot k\_m\right)} \cdot \ell \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* (* (* t k_m) t) (* t k_m))) l))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (((t * k_m) * t) * (t * k_m))) * 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 = (l / (((t * k_m) * t) * (t * k_m))) * l
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / (((t * k_m) * t) * (t * k_m))) * l;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (((t * k_m) * t) * (t * k_m))) * l

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(Float64(Float64(t * k_m) * t) * Float64(t * k_m))) * l)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / (((t * k_m) * t) * (t * k_m))) * l;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {t}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot {t}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {t}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   12. lower-*.f64 51.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

5. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 60.7%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\ell} \]
8. Step-by-step derivation

9. Applied rewrites 63.7%

   \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

10. Final simplification 63.7%

    \[\leadsto \frac{\ell}{\left(\left(t \cdot k\right) \cdot t\right) \cdot \left(t \cdot k\right)} \cdot \ell \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))))