Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.6% → 91.8%

Time: 18.1s

Alternatives: 18

Speedup: 12.5×

• 26.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\tan k\_m}{\ell}\\ t_2 := 2 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;k\_m \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 1.66 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot t\_1\right) \cdot \mathsf{fma}\left(k\_m, k\_m, t\_2\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{t\_2 \cdot \left(\sin k\_m \cdot \tan k\_m\right)}{\ell}, t, \left(k\_m \cdot t\right) \cdot \left(t\_1 \cdot \left(k\_m \cdot \sin k\_m\right)\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (tan k_m) l)) (t_2 (* 2.0 (* t t))))
   (if (<= k_m 5e-74)
     (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
     (if (<= k_m 1.66e+137)
       (* (/ 2.0 (* (* (sin k_m) t_1) (fma k_m k_m t_2))) (/ l t))
       (/
        2.0
        (/
         (fma
          (/ (* t_2 (* (sin k_m) (tan k_m))) l)
          t
          (* (* k_m t) (* t_1 (* k_m (sin k_m)))))
         l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = tan(k_m) / l;
	double t_2 = 2.0 * (t * t);
	double tmp;
	if (k_m <= 5e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else if (k_m <= 1.66e+137) {
		tmp = (2.0 / ((sin(k_m) * t_1) * fma(k_m, k_m, t_2))) * (l / t);
	} else {
		tmp = 2.0 / (fma(((t_2 * (sin(k_m) * tan(k_m))) / l), t, ((k_m * t) * (t_1 * (k_m * sin(k_m))))) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(tan(k_m) / l)
	t_2 = Float64(2.0 * Float64(t * t))
	tmp = 0.0
	if (k_m <= 5e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	elseif (k_m <= 1.66e+137)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k_m) * t_1) * fma(k_m, k_m, t_2))) * Float64(l / t));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(Float64(t_2 * Float64(sin(k_m) * tan(k_m))) / l), t, Float64(Float64(k_m * t) * Float64(t_1 * Float64(k_m * sin(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[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 5e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.66e+137], N[(N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(k$95$m * k$95$m + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$2 * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t + N[(N[(k$95$m * t), $MachinePrecision] * N[(t$95$1 * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.99999999999999998e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 4.99999999999999998e-74 < k < 1.65999999999999991e137

   1. Initial program 52.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 64.4%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 91.8%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 91.7%

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

       1. associate-/r/ N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

       5. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 95.9%

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

   if 1.65999999999999991e137 < k

   1. Initial program 45.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 20.8%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 65.7%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 76.4%

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-commutative N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

   11. Applied egg-rr 84.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 1.66 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{\left(2 \cdot \left(t \cdot t\right)\right) \cdot \left(\sin k \cdot \tan k\right)}{\ell}, t, \left(k \cdot t\right) \cdot \left(\frac{\tan k}{\ell} \cdot \left(k \cdot \sin k\right)\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot \frac{\tan k\_m}{\ell}\\ \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 5.4 \cdot 10^{+215}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k\_m, k\_m \cdot t\_1, 2 \cdot \left(t\_1 \cdot \left(t \cdot t\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \frac{\sin k\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) (/ (tan k_m) l))))
   (if (<= k_m 4.6e-74)
     (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
     (if (<= k_m 5.4e+215)
       (/ 2.0 (/ (* t (fma k_m (* k_m t_1) (* 2.0 (* t_1 (* t t))))) l))
       (/ 2.0 (* k_m (* (* k_m t) (* (tan k_m) (/ (sin k_m) (* l l))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * (tan(k_m) / l);
	double tmp;
	if (k_m <= 4.6e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else if (k_m <= 5.4e+215) {
		tmp = 2.0 / ((t * fma(k_m, (k_m * t_1), (2.0 * (t_1 * (t * t))))) / l);
	} else {
		tmp = 2.0 / (k_m * ((k_m * t) * (tan(k_m) * (sin(k_m) / (l * l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * Float64(tan(k_m) / l))
	tmp = 0.0
	if (k_m <= 4.6e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	elseif (k_m <= 5.4e+215)
		tmp = Float64(2.0 / Float64(Float64(t * fma(k_m, Float64(k_m * t_1), Float64(2.0 * Float64(t_1 * Float64(t * t))))) / l));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(k_m * t) * Float64(tan(k_m) * Float64(sin(k_m) / Float64(l * 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[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 4.6e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.4e+215], N[(2.0 / N[(N[(t * N[(k$95$m * N[(k$95$m * t$95$1), $MachinePrecision] + N[(2.0 * N[(t$95$1 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.59999999999999961e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 4.59999999999999961e-74 < k < 5.4e215

   1. Initial program 51.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 54.3%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 89.1%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 93.5%

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

   if 5.4e215 < k

   1. Initial program 43.8%

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

   3. Taylor expanded in k around 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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-lowering-*.f64 N/A

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

   5. Simplified 19.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 43.0%

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

   8. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 77.2

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

   10. Simplified 77.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+215}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k, k \cdot \left(\sin k \cdot \frac{\tan k}{\ell}\right), 2 \cdot \left(\left(\sin k \cdot \frac{\tan k}{\ell}\right) \cdot \left(t \cdot t\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \frac{\tan k\_m}{\ell}\right) \cdot \mathsf{fma}\left(k\_m, k\_m, 2 \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \frac{\sin k\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.2e-74)
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
   (if (<= k_m 1.05e+139)
     (*
      (/ 2.0 (* (* (sin k_m) (/ (tan k_m) l)) (fma k_m k_m (* 2.0 (* t t)))))
      (/ l t))
     (/ 2.0 (* k_m (* (* k_m t) (* (tan k_m) (/ (sin k_m) (* l l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else if (k_m <= 1.05e+139) {
		tmp = (2.0 / ((sin(k_m) * (tan(k_m) / l)) * fma(k_m, k_m, (2.0 * (t * t))))) * (l / t);
	} else {
		tmp = 2.0 / (k_m * ((k_m * t) * (tan(k_m) * (sin(k_m) / (l * l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	elseif (k_m <= 1.05e+139)
		tmp = Float64(Float64(2.0 / Float64(Float64(sin(k_m) * Float64(tan(k_m) / l)) * fma(k_m, k_m, Float64(2.0 * Float64(t * t))))) * Float64(l / t));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(k_m * t) * Float64(tan(k_m) * Float64(sin(k_m) / Float64(l * l))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.2e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.05e+139], N[(N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.1999999999999999e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 3.1999999999999999e-74 < k < 1.0499999999999999e139

   1. Initial program 52.8%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 64.4%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 91.8%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 91.7%

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

       1. associate-/r/ N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

       5. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 95.9%

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

   if 1.0499999999999999e139 < k

   1. Initial program 45.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-lowering-*.f64 N/A

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

   5. Simplified 28.4%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 45.2%

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

   8. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 78.9

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

   10. Simplified 78.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 3.1 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 13500000:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(2, t\_1, \left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \frac{\sin k\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= k_m 3.1e-74)
     (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
     (if (<= k_m 13500000.0)
       (/
        2.0
        (/
         (*
          t
          (*
           (* k_m k_m)
           (fma
            2.0
            t_1
            (* (* k_m k_m) (fma t_1 0.3333333333333333 (/ 1.0 l))))))
         l))
       (if (<= k_m 4e+127)
         (/ 2.0 (/ (* (* (sin k_m) (tan k_m)) (/ (* t (* k_m k_m)) l)) l))
         (/ 2.0 (* k_m (* (* k_m t) (* (tan k_m) (/ (sin k_m) (* l l)))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (k_m <= 3.1e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else if (k_m <= 13500000.0) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma(2.0, t_1, ((k_m * k_m) * fma(t_1, 0.3333333333333333, (1.0 / l)))))) / l);
	} else if (k_m <= 4e+127) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * ((t * (k_m * k_m)) / l)) / l);
	} else {
		tmp = 2.0 / (k_m * ((k_m * t) * (tan(k_m) * (sin(k_m) / (l * l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (k_m <= 3.1e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	elseif (k_m <= 13500000.0)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(2.0, t_1, Float64(Float64(k_m * k_m) * fma(t_1, 0.3333333333333333, Float64(1.0 / l)))))) / l));
	elseif (k_m <= 4e+127)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64(t * Float64(k_m * k_m)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(k_m * t) * Float64(tan(k_m) * Float64(sin(k_m) / Float64(l * 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]}, If[LessEqual[k$95$m, 3.1e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 13500000.0], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * t$95$1 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e+127], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 3.1000000000000002e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 3.1000000000000002e-74 < k < 1.35e7

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 65.8%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 92.6%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 92.6%

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

   10. Taylor expanded in k around 0

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. /-lowering-/.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-commutative N/A

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

       12. accelerator-lowering-fma.f64 N/A

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

       13. /-lowering-/.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 N/A

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

       16. /-lowering-/.f64 89.4

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

   12. Simplified 89.4%

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

   if 1.35e7 < k < 3.99999999999999982e127

   1. Initial program 55.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 65.6%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 66.0%

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

   7. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 90.4

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

   9. Simplified 90.4%

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

   if 3.99999999999999982e127 < k

   1. Initial program 44.0%

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

   3. Taylor expanded in k around 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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-lowering-*.f64 N/A

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

   5. Simplified 30.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 46.6%

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

   8. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 79.4

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

   10. Simplified 79.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 13500000:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot t}{\ell}, \left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 3.8 \cdot 10^{+127}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\sin k\_m \cdot \frac{\tan k\_m}{\ell}\right) \cdot \mathsf{fma}\left(k\_m, k\_m, 2 \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \frac{\sin k\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.9e-74)
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
   (if (<= k_m 3.8e+127)
     (*
      l
      (/
       2.0
       (* t (* (* (sin k_m) (/ (tan k_m) l)) (fma k_m k_m (* 2.0 (* t t)))))))
     (/ 2.0 (* k_m (* (* k_m t) (* (tan k_m) (/ (sin k_m) (* l l)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.9e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else if (k_m <= 3.8e+127) {
		tmp = l * (2.0 / (t * ((sin(k_m) * (tan(k_m) / l)) * fma(k_m, k_m, (2.0 * (t * t))))));
	} else {
		tmp = 2.0 / (k_m * ((k_m * t) * (tan(k_m) * (sin(k_m) / (l * l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.9e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	elseif (k_m <= 3.8e+127)
		tmp = Float64(l * Float64(2.0 / Float64(t * Float64(Float64(sin(k_m) * Float64(tan(k_m) / l)) * fma(k_m, k_m, Float64(2.0 * Float64(t * t)))))));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(k_m * t) * Float64(tan(k_m) * Float64(sin(k_m) / Float64(l * l))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.9e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.8e+127], N[(l * N[(2.0 / N[(t * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.9e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 2.9e-74 < k < 3.7999999999999998e127

   1. Initial program 53.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 65.7%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 91.6%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 91.6%

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

       1. associate-/r/ N/A

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

       2. *-lowering-*.f64 N/A

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

   11. Applied egg-rr 92.0%

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

   if 3.7999999999999998e127 < k

   1. Initial program 44.0%

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

   3. Taylor expanded in k around 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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-lowering-*.f64 N/A

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

   5. Simplified 30.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 46.6%

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

   8. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 79.4

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

   10. Simplified 79.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+127}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\left(\sin k \cdot \frac{\tan k}{\ell}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{elif}\;k\_m \leq 13500000:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(2, t\_1, \left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \left(\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \frac{\sin k\_m}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= k_m 2.8e-74)
     (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
     (if (<= k_m 13500000.0)
       (/
        2.0
        (/
         (*
          t
          (*
           (* k_m k_m)
           (fma
            2.0
            t_1
            (* (* k_m k_m) (fma t_1 0.3333333333333333 (/ 1.0 l))))))
         l))
       (/ 2.0 (* k_m (* (* k_m t) (* (tan k_m) (/ (sin k_m) (* l l))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (k_m <= 2.8e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else if (k_m <= 13500000.0) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma(2.0, t_1, ((k_m * k_m) * fma(t_1, 0.3333333333333333, (1.0 / l)))))) / l);
	} else {
		tmp = 2.0 / (k_m * ((k_m * t) * (tan(k_m) * (sin(k_m) / (l * l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (k_m <= 2.8e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	elseif (k_m <= 13500000.0)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(2.0, t_1, Float64(Float64(k_m * k_m) * fma(t_1, 0.3333333333333333, Float64(1.0 / l)))))) / l));
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(k_m * t) * Float64(tan(k_m) * Float64(sin(k_m) / Float64(l * 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]}, If[LessEqual[k$95$m, 2.8e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 13500000.0], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * t$95$1 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.79999999999999988e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 2.79999999999999988e-74 < k < 1.35e7

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 65.8%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 92.6%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 92.6%

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

   10. Taylor expanded in k around 0

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. /-lowering-/.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-commutative N/A

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

       12. accelerator-lowering-fma.f64 N/A

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

       13. /-lowering-/.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 N/A

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

       16. /-lowering-/.f64 89.4

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

   12. Simplified 89.4%

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

   if 1.35e7 < k

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. *-lowering-*.f64 N/A

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

   5. Simplified 49.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 60.2%

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

   8. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 81.5

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

   10. Simplified 81.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 13500000:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot t}{\ell}, \left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(2, t\_1, \left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot t} \cdot \frac{\ell \cdot \frac{1}{t}}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= t 1.6e-33)
     (/
      2.0
      (/
       (*
        t
        (*
         (* k_m k_m)
         (fma 2.0 t_1 (* (* k_m k_m) (fma t_1 0.3333333333333333 (/ 1.0 l))))))
       l))
     (* (/ l (* k_m t)) (/ (* l (/ 1.0 t)) (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (t <= 1.6e-33) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma(2.0, t_1, ((k_m * k_m) * fma(t_1, 0.3333333333333333, (1.0 / l)))))) / l);
	} else {
		tmp = (l / (k_m * t)) * ((l * (1.0 / t)) / (k_m * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (t <= 1.6e-33)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(2.0, t_1, Float64(Float64(k_m * k_m) * fma(t_1, 0.3333333333333333, Float64(1.0 / l)))))) / l));
	else
		tmp = Float64(Float64(l / Float64(k_m * t)) * Float64(Float64(l * Float64(1.0 / t)) / Float64(k_m * t)));
	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]}, If[LessEqual[t, 1.6e-33], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * t$95$1 + N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999988e-33

   1. Initial program 45.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 39.0%

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

   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. *-lowering-*.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow-lowering-pow.f64 N/A

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

      7. sin-lowering-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 74.0%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   9. Applied egg-rr 79.3%

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

   10. Taylor expanded in k around 0

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. /-lowering-/.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-commutative N/A

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

       12. accelerator-lowering-fma.f64 N/A

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

       13. /-lowering-/.f64 N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 N/A

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

       16. /-lowering-/.f64 60.7

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

   12. Simplified 60.7%

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

   if 1.59999999999999988e-33 < t

   1. Initial program 78.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 75.2

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

   5. Simplified 75.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 79.7

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

   7. Applied egg-rr 79.7%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 78.4

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

   9. Applied egg-rr 78.4%

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

       1. associate-/r* N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. times-frac N/A

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

       7. associate-/l/ N/A

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

       8. times-frac N/A

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

       9. div-inv N/A

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

       10. associate-*l* N/A

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

       11. times-frac N/A

           \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]

       13. associate-/r* N/A

           \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\color{blue}{\frac{1}{t} \cdot \ell}}{k \cdot t} \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\color{blue}{\frac{1}{t}} \cdot \ell}{k \cdot t} \]

       19. *-lowering-*.f64 88.8

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\frac{1}{t} \cdot \ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 88.8%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot t}{\ell}, \left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\ell \cdot \frac{1}{t}}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-61}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{\ell \cdot \ell}, 0.16666666666666666, \frac{1}{\ell \cdot \ell}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.5e-61)
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
   (/
    2.0
    (*
     t
     (*
      (*
       (* k_m k_m)
       (fma (/ (* k_m k_m) (* l l)) 0.16666666666666666 (/ 1.0 (* l l))))
      (fma 2.0 (* t t) (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-61) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else {
		tmp = 2.0 / (t * (((k_m * k_m) * fma(((k_m * k_m) / (l * l)), 0.16666666666666666, (1.0 / (l * l)))) * fma(2.0, (t * t), (k_m * k_m))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-61)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(k_m * k_m) * fma(Float64(Float64(k_m * k_m) / Float64(l * l)), 0.16666666666666666, Float64(1.0 / Float64(l * l)))) * fma(2.0, Float64(t * t), Float64(k_m * k_m)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.5e-61], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t * t), $MachinePrecision] + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4999999999999999e-61

   1. Initial program 56.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.1

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

   5. Simplified 57.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.2

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

   7. Applied egg-rr 63.2%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.1

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

   9. Applied egg-rr 73.1%

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

   if 2.4999999999999999e-61 < k

   1. Initial program 51.4%

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

   3. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

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

   5. Simplified 77.3%

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

   6. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right)\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{1}{6}} + \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2}}{{\ell}^{2}}, \frac{1}{6}, \frac{1}{{\ell}^{2}}\right)}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{{k}^{2}}{{\ell}^{2}}}, \frac{1}{6}, \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}}, \frac{1}{6}, \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}}, \frac{1}{6}, \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}, \frac{1}{6}, \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}, \frac{1}{6}, \frac{1}{{\ell}^{2}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell \cdot \ell}, \frac{1}{6}, \color{blue}{\frac{1}{{\ell}^{2}}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell \cdot \ell}, \frac{1}{6}, \frac{1}{\color{blue}{\ell \cdot \ell}}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]

      13. *-lowering-*.f64 65.9

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

   8. Simplified 65.9%

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell \cdot \ell}, 0.16666666666666666, \frac{1}{\ell \cdot \ell}\right)\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-61}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell \cdot \ell}, 0.16666666666666666, \frac{1}{\ell \cdot \ell}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{k\_m \cdot k\_m}{\ell \cdot \ell} \cdot \mathsf{fma}\left(2, t \cdot t, k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.2e-74)
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
   (/ 2.0 (* t (* (/ (* k_m k_m) (* l l)) (fma 2.0 (* t t) (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-74) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else {
		tmp = 2.0 / (t * (((k_m * k_m) / (l * l)) * fma(2.0, (t * t), (k_m * k_m))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-74)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(k_m * k_m) / Float64(l * l)) * fma(2.0, Float64(t * t), Float64(k_m * k_m)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.2e-74], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t * t), $MachinePrecision] + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000002e-74

   1. Initial program 57.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 57.0

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

   5. Simplified 57.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 63.1

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

   7. Applied egg-rr 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 73.2

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

   9. Applied egg-rr 73.2%

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

   if 5.2000000000000002e-74 < k

   1. Initial program 49.6%

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

   3. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

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

   5. Simplified 77.0%

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

   6. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.9

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

   8. Simplified 65.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-74}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot t} \cdot \frac{\ell \cdot \frac{1}{t}}{k\_m \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.75e-39)
   (/ 2.0 (* (* k_m k_m) (/ (* t (* k_m k_m)) (* l l))))
   (* (/ l (* k_m t)) (/ (* l (/ 1.0 t)) (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.75e-39) {
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	} else {
		tmp = (l / (k_m * t)) * ((l * (1.0 / t)) / (k_m * t));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.75d-39) then
        tmp = 2.0d0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)))
    else
        tmp = (l / (k_m * t)) * ((l * (1.0d0 / t)) / (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.75e-39) {
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	} else {
		tmp = (l / (k_m * t)) * ((l * (1.0 / t)) / (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.75e-39:
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)))
	else:
		tmp = (l / (k_m * t)) * ((l * (1.0 / t)) / (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.75e-39)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t * Float64(k_m * k_m)) / Float64(l * l))));
	else
		tmp = Float64(Float64(l / Float64(k_m * t)) * Float64(Float64(l * Float64(1.0 / t)) / Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.75e-39)
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	else
		tmp = (l / (k_m * t)) * ((l * (1.0 / t)) / (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.75e-39], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.75000000000000009e-39

   1. Initial program 45.4%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

   5. Simplified 54.3%

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

   6. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 50.6

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

   8. Simplified 50.6%

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

   if 2.75000000000000009e-39 < t

   1. Initial program 77.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 74.3

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 77.4

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

   9. Applied egg-rr 77.4%

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

       1. associate-/r* N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}} \]

       7. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{t}}{t}} \cdot \frac{\ell}{k \cdot t} \]

       8. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{t} \cdot \ell}{t \cdot \left(k \cdot t\right)}} \]

       9. div-inv N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{\ell}{k} \cdot \frac{1}{t}\right)} \cdot \ell}{t \cdot \left(k \cdot t\right)} \]

       10. associate-*l* N/A

           \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \left(\frac{1}{t} \cdot \ell\right)}}{t \cdot \left(k \cdot t\right)} \]

       11. times-frac N/A

           \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]

       13. associate-/r* N/A

           \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\ell}{k \cdot t}} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{k \cdot t}} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \color{blue}{\frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\color{blue}{\frac{1}{t} \cdot \ell}}{k \cdot t} \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\color{blue}{\frac{1}{t}} \cdot \ell}{k \cdot t} \]

       19. *-lowering-*.f64 87.7

           \[\leadsto \frac{\ell}{k \cdot t} \cdot \frac{\frac{1}{t} \cdot \ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 87.7%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\frac{1}{t} \cdot \ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\ell \cdot \frac{1}{t}}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t \cdot \left(k\_m \cdot k\_m\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 9.6e-40)
   (/ 2.0 (* (* k_m k_m) (/ (* t (* k_m k_m)) (* l l))))
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.6e-40) {
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	} else {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 9.6d-40) then
        tmp = 2.0d0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)))
    else
        tmp = l * ((l / (k_m * t)) / (t * (k_m * t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.6e-40) {
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	} else {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 9.6e-40:
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)))
	else:
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 9.6e-40)
		tmp = Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t * Float64(k_m * k_m)) / Float64(l * l))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 9.6e-40)
		tmp = 2.0 / ((k_m * k_m) * ((t * (k_m * k_m)) / (l * l)));
	else
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 9.6e-40], N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.59999999999999965e-40

   1. Initial program 45.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(2, \frac{{t}^{3}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{3}}{{\ell}^{2}}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \color{blue}{\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}}\right)} \]

   5. Simplified 54.3%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(2, \frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell}, \frac{\left(k \cdot k\right) \cdot \left(t \cdot \mathsf{fma}\left(t \cdot t, 0.3333333333333333, 1\right)\right)}{\ell \cdot \ell}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      6. *-lowering-*.f64 50.6

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 50.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}} \]

   if 9.59999999999999965e-40 < t

   1. Initial program 77.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 74.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 78.7

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot k}} \cdot \ell \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot k}} \cdot \ell \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot t}}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{\color{blue}{t \cdot \left(t \cdot k\right)}} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{t \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \ell \]

      9. *-lowering-*.f64 86.4

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{t \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

   9. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.8e-13)
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
   (/ (/ (* l l) t) (* t (* t (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.8e-13) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else {
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	}
	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 <= 3.8d-13) then
        tmp = l * ((l / (k_m * t)) / (t * (k_m * t)))
    else
        tmp = ((l * l) / t) / (t * (t * (k_m * k_m)))
    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 <= 3.8e-13) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else {
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.8e-13:
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)))
	else:
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.8e-13)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(l * l) / t) / Float64(t * Float64(t * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.8e-13)
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	else
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.8e-13], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.8e-13

   1. Initial program 56.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 58.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 64.0

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot k}} \cdot \ell \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot k}} \cdot \ell \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot t}}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{\color{blue}{t \cdot \left(t \cdot k\right)}} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{t \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \ell \]

      9. *-lowering-*.f64 73.6

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{t \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

   9. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}} \cdot \ell \]

   if 3.8e-13 < k

   1. Initial program 49.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 50.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{t}}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. *-lowering-*.f64 59.9

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 240000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k\_m \cdot t}}{t \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 240000000.0)
   (* l (/ (/ l (* k_m t)) (* t (* k_m t))))
   (/ (* l l) (* t (* t (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 240000000.0) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else {
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	}
	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 <= 240000000.0d0) then
        tmp = l * ((l / (k_m * t)) / (t * (k_m * t)))
    else
        tmp = (l * l) / (t * (t * (t * (k_m * k_m))))
    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 <= 240000000.0) {
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	} else {
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 240000000.0:
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)))
	else:
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 240000000.0)
		tmp = Float64(l * Float64(Float64(l / Float64(k_m * t)) / Float64(t * Float64(k_m * t))));
	else
		tmp = Float64(Float64(l * l) / Float64(t * Float64(t * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 240000000.0)
		tmp = l * ((l / (k_m * t)) / (t * (k_m * t)));
	else
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 240000000.0], N[(l * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t * N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e8

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 58.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 64.2

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot k}} \cdot \ell \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot k}} \cdot \ell \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{\color{blue}{k \cdot t}}}{\left(t \cdot t\right) \cdot k} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{\color{blue}{t \cdot \left(t \cdot k\right)}} \cdot \ell \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{t \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \ell \]

      9. *-lowering-*.f64 73.4

         \[\leadsto \frac{\frac{\ell}{k \cdot t}}{t \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

   9. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}} \cdot \ell \]

   if 2.4e8 < k

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 48.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \]

      7. *-lowering-*.f64 57.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \cdot t} \]

   7. Applied egg-rr 57.4%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 240000000:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(t \cdot \left(k\_m \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.5e-159)
   (* l (/ l (* t (* k_m (* t (* k_m t))))))
   (* (/ l t) (/ l (* t (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.5e-159) {
		tmp = l * (l / (t * (k_m * (t * (k_m * t)))));
	} else {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	}
	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 <= 4.5d-159) then
        tmp = l * (l / (t * (k_m * (t * (k_m * t)))))
    else
        tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
    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 <= 4.5e-159) {
		tmp = l * (l / (t * (k_m * (t * (k_m * t)))));
	} else {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.5e-159:
		tmp = l * (l / (t * (k_m * (t * (k_m * t)))))
	else:
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.5e-159)
		tmp = Float64(l * Float64(l / Float64(t * Float64(k_m * Float64(t * Float64(k_m * t))))));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.5e-159)
		tmp = l * (l / (t * (k_m * (t * (k_m * t)))));
	else
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.5e-159], N[(l * N[(l / N[(t * N[(k$95$m * N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.49999999999999989e-159

   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 52.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 59.5

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot t}} \cdot \ell \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot t}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot t} \cdot \ell \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(k \cdot t\right)\right)}\right) \cdot t} \cdot \ell \]

      8. *-lowering-*.f64 67.2

         \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

   9. Applied egg-rr 67.2%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot t}} \cdot \ell \]

   if 4.49999999999999989e-159 < k

   1. Initial program 56.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 61.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. *-lowering-*.f64 69.4

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 15500000:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(t \cdot \left(k\_m \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 15500000.0)
   (* l (/ l (* (* k_m t) (* t (* k_m t)))))
   (/ (* l l) (* t (* t (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 15500000.0) {
		tmp = l * (l / ((k_m * t) * (t * (k_m * t))));
	} else {
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	}
	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 <= 15500000.0d0) then
        tmp = l * (l / ((k_m * t) * (t * (k_m * t))))
    else
        tmp = (l * l) / (t * (t * (t * (k_m * k_m))))
    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 <= 15500000.0) {
		tmp = l * (l / ((k_m * t) * (t * (k_m * t))));
	} else {
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 15500000.0:
		tmp = l * (l / ((k_m * t) * (t * (k_m * t))))
	else:
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 15500000.0)
		tmp = Float64(l * Float64(l / Float64(Float64(k_m * t) * Float64(t * Float64(k_m * t)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t * Float64(t * Float64(t * Float64(k_m * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 15500000.0)
		tmp = l * (l / ((k_m * t) * (t * (k_m * t))));
	else
		tmp = (l * l) / (t * (t * (t * (k_m * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 15500000.0], N[(l * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t * N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55e7

   1. Initial program 56.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 58.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

      6. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

      8. associate-*l* N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

      11. *-lowering-*.f64 64.2

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

      8. *-lowering-*.f64 70.0

         \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

   9. Applied egg-rr 70.0%

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

   if 1.55e7 < k

   1. Initial program 48.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.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. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 48.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \]

      7. *-lowering-*.f64 57.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \cdot t} \]

   7. Applied egg-rr 57.4%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 15500000:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 66.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(t \cdot \left(k\_m \cdot t\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* t (* k_m (* t (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (t * (k_m * (t * (k_m * t)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (t * (k_m * (t * (k_m * t)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (t * (k_m * (t * (k_m * t)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (t * (k_m * (t * (k_m * t)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(t * Float64(k_m * Float64(t * Float64(k_m * t))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (t * (k_m * (t * (k_m * t)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(t * N[(k$95$m * N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. /-lowering-/.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. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 56.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 60.9

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(t \cdot t\right) \cdot k\right) \cdot t\right)}} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot t}} \cdot \ell \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(t \cdot t\right) \cdot k\right)\right) \cdot t}} \cdot \ell \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot t} \cdot \ell \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(k \cdot t\right)\right)}\right) \cdot t} \cdot \ell \]

   8. *-lowering-*.f64 65.8

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

9. Applied egg-rr 65.8%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(k \cdot t\right)\right)\right) \cdot t}} \cdot \ell \]

10. Final simplification 65.8%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(k \cdot t\right)\right)\right)} \]
11. Add Preprocessing

Alternative 17: 62.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* t (* k_m (* k_m (* t t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (t * (k_m * (k_m * (t * t)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (t * (k_m * (k_m * (t * t)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (t * (k_m * (k_m * (t * t)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (t * (k_m * (k_m * (t * t)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(t * Float64(k_m * Float64(k_m * Float64(t * t))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (t * (k_m * (k_m * (t * t)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(t * N[(k$95$m * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. /-lowering-/.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. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 56.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 60.9

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot k\right)} \cdot \ell \]

   4. *-lowering-*.f64 60.0

      \[\leadsto \frac{\ell}{k \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot k\right)} \cdot \ell \]

9. Applied egg-rr 60.0%

   \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k}} \cdot \ell \]

    2. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \ell \]

    3. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot t\right) \cdot t\right) \cdot t\right)} \cdot k} \cdot \ell \]

    4. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot t\right) \cdot k} \cdot \ell \]

    5. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot t\right)\right)\right)} \cdot k} \cdot \ell \]

    6. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(k \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(t \cdot \left(k \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot \left(k \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

    9. *-commutative N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot k\right)} \cdot \ell \]

    10. associate-*l* N/A

        \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot k\right)} \cdot \ell \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot k\right)} \cdot \ell \]

    12. *-lowering-*.f64 61.7

        \[\leadsto \frac{\ell}{t \cdot \left(\left(k \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot k\right)} \cdot \ell \]

11. Applied egg-rr 61.7%

    \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(\left(k \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

12. Final simplification 61.7%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot \left(t \cdot t\right)\right)\right)} \]
13. Add Preprocessing

Alternative 18: 62.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* k_m (* t (* k_m (* t t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (k_m * (t * (k_m * (t * t)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l / (k_m * (t * (k_m * (t * t)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (k_m * (t * (k_m * (t * t)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (k_m * (t * (k_m * (t * t)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(k_m * Float64(t * Float64(k_m * Float64(t * t))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (k_m * (t * (k_m * (t * t)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(k$95$m * N[(t * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. /-lowering-/.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. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 56.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right) \cdot k}} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

   8. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   11. *-lowering-*.f64 60.9

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]

8. Final simplification 60.9%

   \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(t \cdot t\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))