Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 90.1%

Time: 19.6s

Alternatives: 22

Speedup: 12.5×

• 27.5% 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: 55.3% 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: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \tan k\_m \cdot \frac{\sin k\_m}{\ell}\\ \mathbf{if}\;k\_m \leq 1.12 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \left(k\_m \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k\_m \cdot t, k\_m \cdot t\_1, \left(2 \cdot t\right) \cdot \left(t\_1 \cdot \left(t \cdot t\right)\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (tan k_m) (/ (sin k_m) l))))
   (if (<= k_m 1.12e-108)
     (/
      2.0
      (*
       (* t (* (/ t l) (* (tan k_m) (* k_m (/ t l)))))
       (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0)))))
     (/ 2.0 (/ (fma (* k_m t) (* k_m t_1) (* (* 2.0 t) (* t_1 (* t t)))) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = tan(k_m) * (sin(k_m) / l);
	double tmp;
	if (k_m <= 1.12e-108) {
		tmp = 2.0 / ((t * ((t / l) * (tan(k_m) * (k_m * (t / l))))) * (1.0 + (1.0 + pow((k_m / t), 2.0))));
	} else {
		tmp = 2.0 / (fma((k_m * t), (k_m * t_1), ((2.0 * t) * (t_1 * (t * t)))) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(tan(k_m) * Float64(sin(k_m) / l))
	tmp = 0.0
	if (k_m <= 1.12e-108)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(t / l) * Float64(tan(k_m) * Float64(k_m * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(k_m * t), Float64(k_m * t_1), Float64(Float64(2.0 * t) * Float64(t_1 * Float64(t * t)))) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.12e-108], N[(2.0 / N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * t$95$1), $MachinePrecision] + N[(N[(2.0 * t), $MachinePrecision] * N[(t$95$1 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.11999999999999992e-108

   1. Initial program 61.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 17.6

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

   4. Applied egg-rr 17.6%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. /-lowering-/.f64 80.1

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

   9. Simplified 80.1%

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

   if 1.11999999999999992e-108 < k

   1. Initial program 61.4%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 N/A

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

      19. unpow2 N/A

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

      20. frac-times N/A

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

   4. Applied egg-rr 53.4%

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. unpow2 N/A

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

      11. +-commutative N/A

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

      12. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

   9. Simplified 87.5%

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

       1. distribute-lft-in N/A

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

       2. associate-*l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

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

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

   11. Applied egg-rr 95.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(k \cdot t, k \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right), \left(2 \cdot t\right) \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot t\right)\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \tan k\_m \cdot \frac{\sin k\_m}{\ell}\\ \mathbf{if}\;k\_m \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \left(k\_m \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \mathsf{fma}\left(k\_m, k\_m \cdot t\_1, 2 \cdot \left(t\_1 \cdot \left(t \cdot t\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (tan k_m) (/ (sin k_m) l))))
   (if (<= k_m 7.4e-110)
     (/
      2.0
      (*
       (* t (* (/ t l) (* (tan k_m) (* k_m (/ t l)))))
       (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0)))))
     (/ 2.0 (* (/ t l) (fma k_m (* k_m t_1) (* 2.0 (* t_1 (* t t)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = tan(k_m) * (sin(k_m) / l);
	double tmp;
	if (k_m <= 7.4e-110) {
		tmp = 2.0 / ((t * ((t / l) * (tan(k_m) * (k_m * (t / l))))) * (1.0 + (1.0 + pow((k_m / t), 2.0))));
	} else {
		tmp = 2.0 / ((t / l) * fma(k_m, (k_m * t_1), (2.0 * (t_1 * (t * t)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(tan(k_m) * Float64(sin(k_m) / l))
	tmp = 0.0
	if (k_m <= 7.4e-110)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(t / l) * Float64(tan(k_m) * Float64(k_m * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * fma(k_m, Float64(k_m * t_1), Float64(2.0 * Float64(t_1 * Float64(t * t))))));
	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] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 7.4e-110], N[(2.0 / N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * 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]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.40000000000000032e-110

   1. Initial program 61.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 17.6

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

   4. Applied egg-rr 17.6%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. /-lowering-/.f64 80.1

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

   9. Simplified 80.1%

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

   if 7.40000000000000032e-110 < k

   1. Initial program 61.4%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 N/A

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

      19. unpow2 N/A

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

      20. frac-times N/A

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

   4. Applied egg-rr 53.4%

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. unpow2 N/A

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

      11. +-commutative N/A

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

      12. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

   9. Simplified 87.5%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

   11. Applied egg-rr 90.4%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \mathsf{fma}\left(k, k \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right), 2 \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot t\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \tan k\_m \cdot \frac{\sin k\_m}{\ell}\\ \mathbf{if}\;k\_m \leq 7.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \left(k\_m \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \mathsf{fma}\left(k\_m, k\_m \cdot t\_1, 2 \cdot \left(t\_1 \cdot \left(t \cdot t\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (tan k_m) (/ (sin k_m) l))))
   (if (<= k_m 7.8e-106)
     (/
      2.0
      (*
       (* t (* (/ t l) (* (tan k_m) (* k_m (/ t l)))))
       (+ 1.0 (+ 1.0 (pow (/ k_m t) 2.0)))))
     (/ (* 2.0 l) (* t (fma k_m (* k_m t_1) (* 2.0 (* t_1 (* t t)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = tan(k_m) * (sin(k_m) / l);
	double tmp;
	if (k_m <= 7.8e-106) {
		tmp = 2.0 / ((t * ((t / l) * (tan(k_m) * (k_m * (t / l))))) * (1.0 + (1.0 + pow((k_m / t), 2.0))));
	} else {
		tmp = (2.0 * l) / (t * fma(k_m, (k_m * t_1), (2.0 * (t_1 * (t * t)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(tan(k_m) * Float64(sin(k_m) / l))
	tmp = 0.0
	if (k_m <= 7.8e-106)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(t / l) * Float64(tan(k_m) * Float64(k_m * Float64(t / l))))) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t * fma(k_m, Float64(k_m * t_1), Float64(2.0 * Float64(t_1 * Float64(t * t))))));
	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] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 7.8e-106], N[(2.0 / N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / 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]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.80000000000000019e-106

   1. Initial program 61.1%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 17.6

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

   4. Applied egg-rr 17.6%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

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

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

      3. /-lowering-/.f64 80.1

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

   9. Simplified 80.1%

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

   if 7.80000000000000019e-106 < k

   1. Initial program 61.4%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 N/A

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

      19. unpow2 N/A

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

      20. frac-times N/A

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

   4. Applied egg-rr 53.4%

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. unpow2 N/A

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

      11. +-commutative N/A

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

      12. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

   9. Simplified 87.5%

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

       1. associate-/r/ N/A

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

       2. associate-*l/ N/A

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

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

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

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

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

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

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

       6. associate-*l* N/A

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

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

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

   11. Applied egg-rr 91.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \mathsf{fma}\left(k, k \cdot \left(\tan k \cdot \frac{\sin k}{\ell}\right), 2 \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell}\right) \cdot \left(t \cdot t\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot {\sin k\_m}^{2}\right)}{\ell \cdot \cos k\_m}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.8e-76)
   (/ 2.0 (/ (/ (* (* k_m k_m) (* t (pow (sin k_m) 2.0))) (* l (cos k_m))) l))
   (/
    (* 2.0 l)
    (*
     t
     (*
      (/ (* t (* t (sin k_m))) l)
      (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.8e-76) {
		tmp = 2.0 / ((((k_m * k_m) * (t * pow(sin(k_m), 2.0))) / (l * cos(k_m))) / l);
	} else {
		tmp = (2.0 * l) / (t * (((t * (t * sin(k_m))) / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.8e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * Float64(t * (sin(k_m) ^ 2.0))) / Float64(l * cos(k_m))) / l));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t * Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.8e-76], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000001e-76

   1. Initial program 55.1%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 N/A

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

      19. unpow2 N/A

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

      20. frac-times N/A

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

   4. Applied egg-rr 43.8%

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. unpow2 N/A

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

      11. +-commutative N/A

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

      12. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 58.7%

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

   7. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 71.0

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

   9. Simplified 71.0%

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

   if 2.8000000000000001e-76 < t

   1. Initial program 73.4%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 38.2

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

   4. Applied egg-rr 38.2%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 96.3%

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

   8. Applied egg-rr 94.2%

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

Alternative 5: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\frac{{\sin k\_m}^{2} \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 8.5e-115)
   (/ 2.0 (/ (* (pow (sin k_m) 2.0) (* t (* k_m k_m))) (* l (* l (cos k_m)))))
   (/
    (* 2.0 l)
    (*
     t
     (*
      (/ (* t (* t (sin k_m))) l)
      (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8.5e-115) {
		tmp = 2.0 / ((pow(sin(k_m), 2.0) * (t * (k_m * k_m))) / (l * (l * cos(k_m))));
	} else {
		tmp = (2.0 * l) / (t * (((t * (t * sin(k_m))) / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 8.5e-115)
		tmp = Float64(2.0 / Float64(Float64((sin(k_m) ^ 2.0) * Float64(t * Float64(k_m * k_m))) / Float64(l * Float64(l * cos(k_m)))));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t * Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 8.5e-115], N[(2.0 / N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.49999999999999953e-115

   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 t around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 68.3

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

   5. Simplified 68.3%

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

   if 8.49999999999999953e-115 < t

   1. Initial program 73.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 38.5

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

   4. Applied egg-rr 38.5%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 95.0%

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

   8. Applied egg-rr 91.9%

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

Alternative 6: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\cos k\_m \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{{\sin k\_m}^{2} \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 8e-115)
   (/ (* (cos k_m) (* 2.0 (* l l))) (* (pow (sin k_m) 2.0) (* t (* k_m k_m))))
   (/
    (* 2.0 l)
    (*
     t
     (*
      (/ (* t (* t (sin k_m))) l)
      (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 8e-115) {
		tmp = (cos(k_m) * (2.0 * (l * l))) / (pow(sin(k_m), 2.0) * (t * (k_m * k_m)));
	} else {
		tmp = (2.0 * l) / (t * (((t * (t * sin(k_m))) / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 8e-115)
		tmp = Float64(Float64(cos(k_m) * Float64(2.0 * Float64(l * l))) / Float64((sin(k_m) ^ 2.0) * Float64(t * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t * Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 8e-115], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000004e-115

   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 t around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 68.3

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

   5. Simplified 68.3%

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

   if 8.0000000000000004e-115 < t

   1. Initial program 73.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 38.5

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

   4. Applied egg-rr 38.5%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 95.0%

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

   8. Applied egg-rr 91.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-116}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot {\sin k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.5e-116)
   (* (* 2.0 (* l l)) (/ (cos k_m) (* (* k_m k_m) (* t (pow (sin k_m) 2.0)))))
   (/
    (* 2.0 l)
    (*
     t
     (*
      (/ (* t (* t (sin k_m))) l)
      (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.5e-116) {
		tmp = (2.0 * (l * l)) * (cos(k_m) / ((k_m * k_m) * (t * pow(sin(k_m), 2.0))));
	} else {
		tmp = (2.0 * l) / (t * (((t * (t * sin(k_m))) / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.5e-116)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k_m) / Float64(Float64(k_m * k_m) * Float64(t * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t * Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.5e-116], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.4999999999999998e-116

   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. Step-by-step derivation

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 5.6

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

   4. Applied egg-rr 5.6%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

      14. sin-lowering-sin.f64 68.3

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

   9. Simplified 68.3%

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

   if 5.4999999999999998e-116 < t

   1. Initial program 73.6%

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

      1. pow-to-exp N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. div-exp N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. log-lowering-log.f64 38.5

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

   4. Applied egg-rr 38.5%

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. cube-unmult N/A

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

      5. pow-to-exp N/A

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

      6. pow2 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. frac-times N/A

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

      11. associate-*l* N/A

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

      12. associate-/l* N/A

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

      13. associate-*l* N/A

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

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

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

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

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

   6. Applied egg-rr 95.0%

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

   8. Applied egg-rr 91.9%

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

Alternative 8: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ t_2 := t \cdot \sin k\_m\\ \mathbf{if}\;t \leq 9 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, \mathsf{fma}\left(2, t\_1 \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_1\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\tan k\_m \cdot \left(\mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right) \cdot \frac{t \cdot \left(t \cdot t\_2\right)}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \frac{t\_2}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)) (t_2 (* t (sin k_m))))
   (if (<= t 9e-84)
     (/
      2.0
      (/
       (*
        t
        (*
         (* k_m k_m)
         (fma
          (* k_m k_m)
          (fma 2.0 (* t_1 0.16666666666666666) (/ 1.0 l))
          (* 2.0 t_1))))
       l))
     (if (<= t 1.2e+52)
       (/
        2.0
        (*
         (tan k_m)
         (* (fma k_m (/ k_m (* t t)) 2.0) (/ (* t (* t t_2)) (* l l)))))
       (/ 2.0 (* 2.0 (* t (* (/ t l) (* (tan k_m) (/ t_2 l))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double t_2 = t * sin(k_m);
	double tmp;
	if (t <= 9e-84) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma((k_m * k_m), fma(2.0, (t_1 * 0.16666666666666666), (1.0 / l)), (2.0 * t_1)))) / l);
	} else if (t <= 1.2e+52) {
		tmp = 2.0 / (tan(k_m) * (fma(k_m, (k_m / (t * t)), 2.0) * ((t * (t * t_2)) / (l * l))));
	} else {
		tmp = 2.0 / (2.0 * (t * ((t / l) * (tan(k_m) * (t_2 / l)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	t_2 = Float64(t * sin(k_m))
	tmp = 0.0
	if (t <= 9e-84)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(Float64(k_m * k_m), fma(2.0, Float64(t_1 * 0.16666666666666666), Float64(1.0 / l)), Float64(2.0 * t_1)))) / l));
	elseif (t <= 1.2e+52)
		tmp = Float64(2.0 / Float64(tan(k_m) * Float64(fma(k_m, Float64(k_m / Float64(t * t)), 2.0) * Float64(Float64(t * Float64(t * t_2)) / Float64(l * l)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(t * Float64(Float64(t / l) * Float64(tan(k_m) * Float64(t_2 / l))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 9e-84], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$1 * 0.16666666666666666), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e+52], N[(2.0 / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(t * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.00000000000000031e-84

   1. Initial program 54.6%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 N/A

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

      19. unpow2 N/A

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

      20. frac-times N/A

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

   4. Applied egg-rr 43.1%

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. unpow2 N/A

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

      11. +-commutative N/A

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

      12. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 58.7%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*r/ N/A

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

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

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

   9. Simplified 84.9%

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

   10. Taylor expanded in k around 0

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

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

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

       2. unpow2 N/A

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

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

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

       4. +-commutative N/A

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. distribute-rgt-out-- N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

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

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

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

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       13. unpow2 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

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

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

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

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \color{blue}{\frac{1}{\ell}}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

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

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), \color{blue}{2 \cdot \frac{{t}^{2}}{\ell}}\right)\right)}{\ell}} \]

   12. Simplified 72.4%

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

   if 9.00000000000000031e-84 < t < 1.2e52

   1. Initial program 82.3%

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

      1. associate-*l* N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

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

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

      6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 N/A

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

      19. unpow2 N/A

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

      20. frac-times N/A

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

   4. Applied egg-rr 85.9%

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. associate-+r+ N/A

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

      10. times-frac N/A

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

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(1 + \left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right) \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \tan k\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \]

   6. Applied egg-rr 79.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \frac{t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \ell}\right) \cdot \tan k}} \]

   if 1.2e52 < t

   1. Initial program 70.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. log-lowering-log.f64 38.5

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 38.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\color{blue}{\log t \cdot 3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-unmult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Simplified 99.7%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \frac{t \cdot \left(t \cdot \left(t \cdot \sin k\right)\right)}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;t \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, \mathsf{fma}\left(2, t\_1 \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_1\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+52}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\sin k\_m \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \frac{t \cdot \sin k\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= t 8e-84)
     (/
      2.0
      (/
       (*
        t
        (*
         (* k_m k_m)
         (fma
          (* k_m k_m)
          (fma 2.0 (* t_1 0.16666666666666666) (/ 1.0 l))
          (* 2.0 t_1))))
       l))
     (if (<= t 1.2e+52)
       (*
        (* l l)
        (/
         (/ 2.0 (* t (* t t)))
         (* (sin k_m) (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0)))))
       (/
        2.0
        (* 2.0 (* t (* (/ t l) (* (tan k_m) (/ (* t (sin k_m)) l))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (t <= 8e-84) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma((k_m * k_m), fma(2.0, (t_1 * 0.16666666666666666), (1.0 / l)), (2.0 * t_1)))) / l);
	} else if (t <= 1.2e+52) {
		tmp = (l * l) * ((2.0 / (t * (t * t))) / (sin(k_m) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	} else {
		tmp = 2.0 / (2.0 * (t * ((t / l) * (tan(k_m) * ((t * sin(k_m)) / 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 (t <= 8e-84)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(Float64(k_m * k_m), fma(2.0, Float64(t_1 * 0.16666666666666666), Float64(1.0 / l)), Float64(2.0 * t_1)))) / l));
	elseif (t <= 1.2e+52)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / Float64(t * Float64(t * t))) / Float64(sin(k_m) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(t * Float64(Float64(t / l) * Float64(tan(k_m) * Float64(Float64(t * 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[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 8e-84], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$1 * 0.16666666666666666), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e+52], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 8.0000000000000003e-84

   1. Initial program 54.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\color{blue}{\sin k}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]

      16. associate-+r+ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      20. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right)\right)\right)} \]

   4. Applied egg-rr 43.1%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      5. frac-times N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + 1\right)} + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   6. Applied egg-rr 58.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

   9. Simplified 84.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \cos k}\right)}}{\ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       4. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right) + 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       6. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\mathsf{fma}\left(2, \frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}, \frac{1}{\ell}\right)}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       9. distribute-rgt-out-- N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{-1}{3} - \frac{-1}{2}\right)}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       13. unpow2 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \color{blue}{\frac{1}{\ell}}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), \color{blue}{2 \cdot \frac{{t}^{2}}{\ell}}\right)\right)}{\ell}} \]

   12. Simplified 72.4%

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}}{\ell}} \]

   if 8.0000000000000003e-84 < t < 1.2e52

   1. Initial program 82.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. log-lowering-log.f64 37.9

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 37.9%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]

   if 1.2e52 < t

   1. Initial program 70.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. log-lowering-log.f64 38.5

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 38.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\color{blue}{\log t \cdot 3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-unmult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Simplified 99.7%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+52}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t \cdot \left(t \cdot t\right)}}{\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;t \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, \mathsf{fma}\left(2, t\_1 \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_1\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\_m\right)}{\ell} \cdot \left(\tan k\_m \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= t 2e-141)
     (/
      2.0
      (/
       (*
        t
        (*
         (* k_m k_m)
         (fma
          (* k_m k_m)
          (fma 2.0 (* t_1 0.16666666666666666) (/ 1.0 l))
          (* 2.0 t_1))))
       l))
     (/
      (* 2.0 l)
      (*
       t
       (*
        (/ (* t (* t (sin k_m))) l)
        (* (tan k_m) (fma k_m (/ k_m (* t t)) 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (t <= 2e-141) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma((k_m * k_m), fma(2.0, (t_1 * 0.16666666666666666), (1.0 / l)), (2.0 * t_1)))) / l);
	} else {
		tmp = (2.0 * l) / (t * (((t * (t * sin(k_m))) / l) * (tan(k_m) * fma(k_m, (k_m / (t * t)), 2.0))));
	}
	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 <= 2e-141)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(Float64(k_m * k_m), fma(2.0, Float64(t_1 * 0.16666666666666666), Float64(1.0 / l)), Float64(2.0 * t_1)))) / l));
	else
		tmp = Float64(Float64(2.0 * l) / Float64(t * Float64(Float64(Float64(t * Float64(t * sin(k_m))) / l) * Float64(tan(k_m) * fma(k_m, Float64(k_m / Float64(t * t)), 2.0)))));
	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, 2e-141], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$1 * 0.16666666666666666), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(N[(N[(t * N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m / N[(t * t), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e-141

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\color{blue}{\sin k}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]

      16. associate-+r+ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      20. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right)\right)\right)} \]

   4. Applied egg-rr 41.6%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      5. frac-times N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + 1\right)} + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   6. Applied egg-rr 57.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

   9. Simplified 84.9%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \cos k}\right)}}{\ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       4. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right) + 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       6. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\mathsf{fma}\left(2, \frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}, \frac{1}{\ell}\right)}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       9. distribute-rgt-out-- N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{-1}{3} - \frac{-1}{2}\right)}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       13. unpow2 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \color{blue}{\frac{1}{\ell}}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), \color{blue}{2 \cdot \frac{{t}^{2}}{\ell}}\right)\right)}{\ell}} \]

   12. Simplified 72.6%

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}}{\ell}} \]

   if 2.0000000000000001e-141 < t

   1. Initial program 72.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. log-lowering-log.f64 37.3

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 37.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\color{blue}{\log t \cdot 3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-unmult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 94.1%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied egg-rr 91.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k\_m \cdot \frac{t \cdot \sin k\_m}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, \mathsf{fma}\left(2, t\_1 \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_1\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= k_m 1.65e-94)
     (/ 2.0 (* 2.0 (* t (* (/ t l) (* (tan k_m) (/ (* t (sin k_m)) l))))))
     (/
      2.0
      (/
       (*
        t
        (*
         (* k_m k_m)
         (fma
          (* k_m k_m)
          (fma 2.0 (* t_1 0.16666666666666666) (/ 1.0 l))
          (* 2.0 t_1))))
       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 <= 1.65e-94) {
		tmp = 2.0 / (2.0 * (t * ((t / l) * (tan(k_m) * ((t * sin(k_m)) / l)))));
	} else {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma((k_m * k_m), fma(2.0, (t_1 * 0.16666666666666666), (1.0 / l)), (2.0 * t_1)))) / 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 <= 1.65e-94)
		tmp = Float64(2.0 / Float64(2.0 * Float64(t * Float64(Float64(t / l) * Float64(tan(k_m) * Float64(Float64(t * sin(k_m)) / l))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(Float64(k_m * k_m), fma(2.0, Float64(t_1 * 0.16666666666666666), Float64(1.0 / l)), Float64(2.0 * t_1)))) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.65e-94], N[(2.0 / N[(2.0 * N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$1 * 0.16666666666666666), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6500000000000001e-94

   1. Initial program 61.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{e^{\log t \cdot 3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\log t \cdot 3}}{\color{blue}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{\color{blue}{3 \cdot \log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \color{blue}{\log t} - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. log-lowering-log.f64 17.1

          \[\leadsto \frac{2}{\left(\left(e^{3 \cdot \log t - \color{blue}{\log \ell} \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 17.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{3 \cdot \log t - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. div-exp N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{e^{3 \cdot \log t}}{e^{\log \ell \cdot 2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{e^{\color{blue}{\log t \cdot 3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-unmult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{e^{\log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. pow-to-exp N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{{\ell}^{2}}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. frac-times N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \tan k\right)\right)}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied egg-rr 86.1%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]
   8. Step-by-step derivation

   9. Simplified 76.5%

      \[\leadsto \frac{2}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \color{blue}{2}} \]

   if 1.6500000000000001e-94 < k

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\color{blue}{\sin k}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]

      16. associate-+r+ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      20. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right)\right)\right)} \]

   4. Applied egg-rr 52.5%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      5. frac-times N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + 1\right)} + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   6. Applied egg-rr 67.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

   9. Simplified 87.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \cos k}\right)}}{\ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       4. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right) + 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       6. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\mathsf{fma}\left(2, \frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}, \frac{1}{\ell}\right)}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       9. distribute-rgt-out-- N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{-1}{3} - \frac{-1}{2}\right)}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       13. unpow2 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \color{blue}{\frac{1}{\ell}}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), \color{blue}{2 \cdot \frac{{t}^{2}}{\ell}}\right)\right)}{\ell}} \]

   12. Simplified 74.0%

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;t \leq 2.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, \mathsf{fma}\left(2, t\_1 \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot t\_1\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+143}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= t 2.9e-84)
     (/
      2.0
      (/
       (*
        t
        (*
         (* k_m k_m)
         (fma
          (* k_m k_m)
          (fma 2.0 (* t_1 0.16666666666666666) (/ 1.0 l))
          (* 2.0 t_1))))
       l))
     (if (<= t 1e+143)
       (* (/ l (* k_m (* t t))) (/ l (* k_m t)))
       (* (/ l t) (/ l (* t (* k_m (* 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 <= 2.9e-84) {
		tmp = 2.0 / ((t * ((k_m * k_m) * fma((k_m * k_m), fma(2.0, (t_1 * 0.16666666666666666), (1.0 / l)), (2.0 * t_1)))) / l);
	} else if (t <= 1e+143) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (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 <= 2.9e-84)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * fma(Float64(k_m * k_m), fma(2.0, Float64(t_1 * 0.16666666666666666), Float64(1.0 / l)), Float64(2.0 * t_1)))) / l));
	elseif (t <= 1e+143)
		tmp = Float64(Float64(l / Float64(k_m * Float64(t * t))) * Float64(l / Float64(k_m * t)));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(k_m * 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, 2.9e-84], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$1 * 0.16666666666666666), $MachinePrecision] + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+143], N[(N[(l / N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.90000000000000019e-84

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\color{blue}{\sin k}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]

      16. associate-+r+ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      20. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right)\right)\right)} \]

   4. Applied egg-rr 42.8%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      5. frac-times N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + 1\right)} + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   6. Applied egg-rr 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]
   8. 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. +-commutative N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{{k}^{2} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} + 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}}{\ell}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{\color{blue}{{\sin k}^{2}}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\color{blue}{\sin k}}^{2}}{\ell \cdot \cos k}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\color{blue}{\ell \cdot \cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \color{blue}{\cos k}}, 2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}\right)}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \color{blue}{\frac{2 \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\right)}{\ell}} \]

   9. Simplified 84.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \mathsf{fma}\left(k \cdot k, \frac{{\sin k}^{2}}{\ell \cdot \cos k}, \frac{2 \cdot \left(t \cdot \left(t \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \cos k}\right)}}{\ell}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}}{\ell}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + {k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right)\right)\right)}{\ell}} \]

       4. +-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}\right) + 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{2}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)}\right)}{\ell}} \]

       6. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{k \cdot k}, 2 \cdot \left(\frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{1}{\ell}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \color{blue}{\mathsf{fma}\left(2, \frac{-1}{3} \cdot \frac{{t}^{2}}{\ell} - \frac{-1}{2} \cdot \frac{{t}^{2}}{\ell}, \frac{1}{\ell}\right)}, 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       9. distribute-rgt-out-- N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{-1}{3} - \frac{-1}{2}\right)}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       10. metadata-eval N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{{t}^{2}}{\ell} \cdot \color{blue}{\frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell} \cdot \frac{1}{6}}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \color{blue}{\frac{{t}^{2}}{\ell}} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       13. unpow2 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       15. /-lowering-/.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \color{blue}{\frac{1}{\ell}}\right), 2 \cdot \frac{{t}^{2}}{\ell}\right)\right)}{\ell}} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot \frac{1}{6}, \frac{1}{\ell}\right), \color{blue}{2 \cdot \frac{{t}^{2}}{\ell}}\right)\right)}{\ell}} \]

   12. Simplified 72.2%

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \mathsf{fma}\left(2, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666, \frac{1}{\ell}\right), 2 \cdot \frac{t \cdot t}{\ell}\right)\right)}}{\ell}} \]

   if 2.90000000000000019e-84 < t < 1e143

   1. Initial program 76.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 62.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 70.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 77.1

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 77.1%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       9. *-lowering-*.f64 82.1

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 1e143 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 64.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 71.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 95.3

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq 3.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, 1, t\_1 \cdot 0.3333333333333333\right), \frac{k\_m \cdot k\_m}{\ell}, \frac{2 \cdot t\_1}{\ell}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (* t t))))
   (if (<= t 3.5e-84)
     (/
      2.0
      (/
       (*
        (* k_m k_m)
        (fma
         (fma t 1.0 (* t_1 0.3333333333333333))
         (/ (* k_m k_m) l)
         (/ (* 2.0 t_1) l)))
       l))
     (if (<= t 2e+141)
       (* (/ l (* k_m (* t t))) (/ l (* k_m t)))
       (* (/ l t) (/ l (* t (* k_m (* k_m t)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * (t * t);
	double tmp;
	if (t <= 3.5e-84) {
		tmp = 2.0 / (((k_m * k_m) * fma(fma(t, 1.0, (t_1 * 0.3333333333333333)), ((k_m * k_m) / l), ((2.0 * t_1) / l))) / l);
	} else if (t <= 2e+141) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * Float64(t * t))
	tmp = 0.0
	if (t <= 3.5e-84)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * fma(fma(t, 1.0, Float64(t_1 * 0.3333333333333333)), Float64(Float64(k_m * k_m) / l), Float64(Float64(2.0 * t_1) / l))) / l));
	elseif (t <= 2e+141)
		tmp = Float64(Float64(l / Float64(k_m * Float64(t * t))) * Float64(l / Float64(k_m * t)));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(k_m * 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[(t * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 3.5e-84], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * 1.0 + N[(t$95$1 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] + N[(N[(2.0 * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e+141], N[(N[(l / N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.5000000000000001e-84

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{\sin k}{\ell \cdot \ell}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{t}^{3} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      6. cube-mult N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\color{blue}{\frac{\sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\color{blue}{\sin k}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]

      16. associate-+r+ N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

      19. unpow2 N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      20. frac-times N/A

          \[\leadsto \frac{2}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right)\right)\right)} \]

   4. Applied egg-rr 42.8%

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      5. frac-times N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \left(\tan k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(\color{blue}{\left(1 + 1\right)} + \frac{k \cdot k}{t \cdot t}\right)\right)} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\right)} \]

      9. times-frac N/A

         \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \left(1 + \left(1 + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{2}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}} \]

   6. Applied egg-rr 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}}{\ell}} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}}{\ell}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}{\ell}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)}{\ell}} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell} + 2 \cdot \frac{{t}^{3}}{\ell}\right)}}{\ell}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\frac{\color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot {k}^{2}}}{\ell} + 2 \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(\color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{\ell}} + 2 \cdot \frac{{t}^{3}}{\ell}\right)}{\ell}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right), \frac{{k}^{2}}{\ell}, 2 \cdot \frac{{t}^{3}}{\ell}\right)}}{\ell}} \]

   9. Simplified 71.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t, 1, \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.3333333333333333\right), \frac{k \cdot k}{\ell}, \frac{2 \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}\right)}}{\ell}} \]

   if 3.5000000000000001e-84 < t < 2.00000000000000003e141

   1. Initial program 76.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 62.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 62.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 70.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 77.1

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 77.1%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       9. *-lowering-*.f64 82.1

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 82.1%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 2.00000000000000003e141 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 64.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 71.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 95.3

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 64.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{+143}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5e-91)
   (/ (/ (* l l) t) (* t (* t (* k_m k_m))))
   (if (<= t 1.66e+143)
     (* (/ l (* k_m (* t t))) (/ l (* k_m t)))
     (* (/ l t) (/ l (* t (* k_m (* k_m t))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5e-91) {
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	} else if (t <= 1.66e+143) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (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 <= 5d-91) then
        tmp = ((l * l) / t) / (t * (t * (k_m * k_m)))
    else if (t <= 1.66d+143) then
        tmp = (l / (k_m * (t * t))) * (l / (k_m * t))
    else
        tmp = (l / t) * (l / (t * (k_m * (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 <= 5e-91) {
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	} else if (t <= 1.66e+143) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 5e-91:
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)))
	elif t <= 1.66e+143:
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t))
	else:
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5e-91)
		tmp = Float64(Float64(Float64(l * l) / t) / Float64(t * Float64(t * Float64(k_m * k_m))));
	elseif (t <= 1.66e+143)
		tmp = Float64(Float64(l / Float64(k_m * Float64(t * t))) * Float64(l / Float64(k_m * t)));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(k_m * 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 <= 5e-91)
		tmp = ((l * l) / t) / (t * (t * (k_m * k_m)));
	elseif (t <= 1.66e+143)
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	else
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5e-91], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.66e+143], N[(N[(l / N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.99999999999999997e-91

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 55.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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. 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 64.6

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   if 4.99999999999999997e-91 < t < 1.66000000000000007e143

   1. Initial program 74.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 67.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 73.2

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 73.2%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       9. *-lowering-*.f64 77.8

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 77.8%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 1.66000000000000007e143 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 64.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 71.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 95.3

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 66.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)\\ \mathbf{if}\;t \leq 4.15 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{t\_1}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* t (* k_m (* k_m t)))))
   (if (<= t 4.15e-106)
     (/ (/ (* l l) t) t_1)
     (if (<= t 4.2e+141)
       (* (/ l (* k_m (* t t))) (/ l (* k_m t)))
       (* (/ l t) (/ l t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * (k_m * t));
	double tmp;
	if (t <= 4.15e-106) {
		tmp = ((l * l) / t) / t_1;
	} else if (t <= 4.2e+141) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / t_1);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (k_m * (k_m * t))
    if (t <= 4.15d-106) then
        tmp = ((l * l) / t) / t_1
    else if (t <= 4.2d+141) then
        tmp = (l / (k_m * (t * t))) * (l / (k_m * t))
    else
        tmp = (l / t) * (l / t_1)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = t * (k_m * (k_m * t));
	double tmp;
	if (t <= 4.15e-106) {
		tmp = ((l * l) / t) / t_1;
	} else if (t <= 4.2e+141) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / t_1);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = t * (k_m * (k_m * t))
	tmp = 0
	if t <= 4.15e-106:
		tmp = ((l * l) / t) / t_1
	elif t <= 4.2e+141:
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t))
	else:
		tmp = (l / t) * (l / t_1)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t * Float64(k_m * Float64(k_m * t)))
	tmp = 0.0
	if (t <= 4.15e-106)
		tmp = Float64(Float64(Float64(l * l) / t) / t_1);
	elseif (t <= 4.2e+141)
		tmp = Float64(Float64(l / Float64(k_m * Float64(t * t))) * Float64(l / Float64(k_m * t)));
	else
		tmp = Float64(Float64(l / t) * Float64(l / t_1));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = t * (k_m * (k_m * t));
	tmp = 0.0;
	if (t <= 4.15e-106)
		tmp = ((l * l) / t) / t_1;
	elseif (t <= 4.2e+141)
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	else
		tmp = (l / t) * (l / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 4.15e-106], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 4.2e+141], N[(N[(l / N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.15000000000000023e-106

   1. Initial program 53.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 60.8

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 60.8%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{t}}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 65.0

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]

   if 4.15000000000000023e-106 < t < 4.1999999999999997e141

   1. Initial program 76.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 62.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 69.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 74.5

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 74.5%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       9. *-lowering-*.f64 78.7

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 4.1999999999999997e141 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 64.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 71.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 95.3

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 67.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.4e-91)
   (* (/ l t) (/ l (* t (* t (* k_m k_m)))))
   (if (<= t 3.6e+141)
     (* (/ l (* k_m (* t t))) (/ l (* k_m t)))
     (* (/ l t) (/ l (* t (* k_m (* k_m t))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.4e-91) {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	} else if (t <= 3.6e+141) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (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 <= 4.4d-91) then
        tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
    else if (t <= 3.6d+141) then
        tmp = (l / (k_m * (t * t))) * (l / (k_m * t))
    else
        tmp = (l / t) * (l / (t * (k_m * (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 <= 4.4e-91) {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	} else if (t <= 3.6e+141) {
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.4e-91:
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
	elif t <= 3.6e+141:
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t))
	else:
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.4e-91)
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(t * Float64(k_m * k_m)))));
	elseif (t <= 3.6e+141)
		tmp = Float64(Float64(l / Float64(k_m * Float64(t * t))) * Float64(l / Float64(k_m * t)));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(k_m * 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 <= 4.4e-91)
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	elseif (t <= 3.6e+141)
		tmp = (l / (k_m * (t * t))) * (l / (k_m * t));
	else
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.4e-91], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e+141], N[(N[(l / N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.4000000000000002e-91

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 55.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 55.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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. 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 67.0

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   if 4.4000000000000002e-91 < t < 3.6000000000000001e141

   1. Initial program 74.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 67.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 73.2

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 73.2%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       9. *-lowering-*.f64 77.8

          \[\leadsto \frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{\color{blue}{k \cdot t}} \]

   11. Applied egg-rr 77.8%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

   if 3.6000000000000001e141 < t

   1. Initial program 71.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 64.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 71.5

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 95.3

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.1e-160)
   (* (/ l t) (/ l (* t (* t (* k_m k_m)))))
   (* (/ l t) (/ l (* t (* k_m (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.1e-160) {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (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 <= 2.1d-160) then
        tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
    else
        tmp = (l / t) * (l / (t * (k_m * (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 <= 2.1e-160) {
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	} else {
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.1e-160:
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))))
	else:
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.1e-160)
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(t * Float64(k_m * k_m)))));
	else
		tmp = Float64(Float64(l / t) * Float64(l / Float64(t * Float64(k_m * 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 <= 2.1e-160)
		tmp = (l / t) * (l / (t * (t * (k_m * k_m))));
	else
		tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.1e-160], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e-160

   1. Initial program 54.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 55.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      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 66.8

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   if 2.1e-160 < t

   1. Initial program 72.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-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 61.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 69.4

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 69.4%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      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. associate-*r* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

      12. *-lowering-*.f64 81.6

          \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   9. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 68.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.5 \cdot 10^{+185}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(t \cdot \left(k\_m \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.5e+185)
   (* 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 <= 5.5e+185) {
		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 <= 5.5d+185) 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 <= 5.5e+185) {
		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 <= 5.5e+185:
		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 <= 5.5e+185)
		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 <= 5.5e+185)
		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, 5.5e+185], 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 < 5.4999999999999996e185

   1. Initial program 62.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 58.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 65.7

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 69.6

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 69.6%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

       3. *-lowering-*.f64 73.1

          \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

   11. Applied egg-rr 73.1%

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

   if 5.4999999999999996e185 < 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 48.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 48.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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. *-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 68.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \cdot t} \]

   7. Applied egg-rr 68.5%

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{+185}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l t) (/ l (* t (* k_m (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / t) * (l / (t * (k_m * (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 / t) * (l / (t * (k_m * (k_m * t))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / t) * (l / (t * (k_m * (k_m * t))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / t) * (l / (t * (k_m * (k_m * t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / t) * Float64(l / Float64(t * Float64(k_m * Float64(k_m * t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / t) * (l / (t * (k_m * (k_m * t))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / t), $MachinePrecision] * N[(l / N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 57.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
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

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 64.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

7. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   2. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   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. associate-*r* N/A

      \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

   12. *-lowering-*.f64 72.8

       \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)} \]

9. Applied egg-rr 72.8%

   \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \]
10. Add Preprocessing

Alternative 20: 65.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* t (* t (* k_m (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (t * (t * (k_m * (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 * (t * (k_m * (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 * (t * (k_m * (k_m * t)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (t * (t * (k_m * (k_m * t)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(t * Float64(t * Float64(k_m * Float64(k_m * t))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (t * (t * (k_m * (k_m * t)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(t * N[(t * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 57.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
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

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 64.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

7. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \cdot \ell \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right) \cdot t}} \cdot \ell \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}\right) \cdot t} \cdot \ell \]

   6. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(k \cdot \left(t \cdot k\right)\right)}\right) \cdot t} \cdot \ell \]

   8. *-commutative N/A

      \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

   9. *-lowering-*.f64 70.8

      \[\leadsto \frac{\ell}{\left(t \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

9. Applied egg-rr 70.8%

   \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot \left(k \cdot t\right)\right)\right) \cdot t}} \cdot \ell \]

10. Final simplification 70.8%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)} \]
11. Add Preprocessing

Alternative 21: 67.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(t \cdot \left(k\_m \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* (* k_m t) (* t (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / ((k_m * t) * (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 / ((k_m * t) * (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 / ((k_m * t) * (t * (k_m * t))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / ((k_m * t) * (t * (k_m * t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(Float64(k_m * t) * Float64(t * Float64(k_m * t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / ((k_m * t) * (t * (k_m * t))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(t * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 57.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
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

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 64.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

7. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 67.5

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

9. Applied egg-rr 67.5%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
10. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

    3. *-lowering-*.f64 70.6

       \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)} \cdot \ell \]

11. Applied egg-rr 70.6%

    \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}} \cdot \ell \]

12. Final simplification 70.6%

    \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(t \cdot \left(k \cdot t\right)\right)} \]
13. Add Preprocessing

Alternative 22: 63.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(k\_m \cdot \left(t \cdot t\right)\right)} \end{array} \]

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(Float64(k_m * 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[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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 57.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 57.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
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

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 64.5

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

7. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 67.5

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

9. Applied egg-rr 67.5%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

10. Final simplification 67.5%

    \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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))))