Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.3% → 94.6%

Time: 18.5s

Alternatives: 17

Speedup: 13.2×

• 39.0% 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: 36.3% 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: 94.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.039:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \left(\frac{2}{t} \cdot \frac{\ell}{k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.039)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (*
    (/ (* l (cos k_m)) k_m)
    (* (/ 2.0 t) (/ l (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.039) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = ((l * cos(k_m)) / k_m) * ((2.0 / t) * (l / (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.039)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(2.0 / t) * Float64(l / Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.039], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] * N[(l / N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0389999999999999999

   1. Initial program 41.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 0.0389999999999999999 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.4

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

   5. Simplified 72.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.5

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

   7. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.5

         \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot t} \]

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 90.5

          \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), \color{blue}{-0.5}, 0.5\right)\right) \cdot t} \]

   9. Applied egg-rr 90.5%

      \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t}} \]
   10. Step-by-step derivation

       1. count-2 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. times-frac N/A

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

       8. lift-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 99.2

           \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \left(\frac{2}{t} \cdot \color{blue}{\frac{\ell}{k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}}\right) \]

   11. Applied egg-rr 99.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.039:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \left(\frac{2}{t} \cdot \frac{\ell}{k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.039:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos k\_m \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell + \ell}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.039)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (*
    (* (cos k_m) (/ l k_m))
    (/ (+ l l) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.039) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = (cos(k_m) * (l / k_m)) * ((l + l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m * t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.039)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(cos(k_m) * Float64(l / k_m)) * Float64(Float64(l + l) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m * t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.039], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0389999999999999999

   1. Initial program 41.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 0.0389999999999999999 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.4

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

   5. Simplified 72.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.5

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

   7. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.5

         \[\leadsto \left(\color{blue}{\frac{\ell}{k}} \cdot \cos k\right) \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \]

   9. Applied egg-rr 90.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.039:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\cos k \cdot \frac{\ell}{k}\right) \cdot \frac{\ell + \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.039:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(k\_m \cdot t\right)} \cdot \left(\ell \cdot \frac{\cos k\_m}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.039)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (*
    (/ (+ l l) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* k_m t)))
    (* l (/ (cos k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.039) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = ((l + l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m * t))) * (l * (cos(k_m) / k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.039)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l + l) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m * t))) * Float64(l * Float64(cos(k_m) / k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.039], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l + l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0389999999999999999

   1. Initial program 41.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 0.0389999999999999999 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.4

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

   5. Simplified 72.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.5

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

   7. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.5

         \[\leadsto \left(\ell \cdot \color{blue}{\frac{\cos k}{k}}\right) \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \]

   9. Applied egg-rr 90.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.039:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\right)} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.039:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\cos k\_m}{k\_m}\right) \cdot \frac{\ell + \ell}{t \cdot \left(k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.039)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (*
    (* l (/ (cos k_m) k_m))
    (/ (+ l l) (* t (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.039) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = (l * (cos(k_m) / k_m)) * ((l + l) / (t * (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.039)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l * Float64(cos(k_m) / k_m)) * Float64(Float64(l + l) / Float64(t * Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.039], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0389999999999999999

   1. Initial program 41.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 0.0389999999999999999 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.4

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

   5. Simplified 72.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.5

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

   7. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.5

         \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot t} \]

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 90.5

          \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), \color{blue}{-0.5}, 0.5\right)\right) \cdot t} \]

   9. Applied egg-rr 90.5%

      \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t}} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 90.4

          \[\leadsto \left(\ell \cdot \color{blue}{\frac{\cos k}{k}}\right) \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t} \]

   11. Applied egg-rr 90.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.039:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\cos k}{k}\right) \cdot \frac{\ell + \ell}{t \cdot \left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.039:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos k\_m\right) \cdot \frac{\ell + \ell}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.039)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (*
    (* l (cos k_m))
    (/ (+ l l) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* (* k_m k_m) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.039) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = (l * cos(k_m)) * ((l + l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * ((k_m * k_m) * t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.039)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l * cos(k_m)) * Float64(Float64(l + l) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(Float64(k_m * k_m) * t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.039], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0389999999999999999

   1. Initial program 41.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 53.2%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 0.0389999999999999999 < k

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.4

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

   5. Simplified 72.4%

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

   7. Applied egg-rr 76.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.039:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos k\right) \cdot \frac{\ell + \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-97}:\\ \;\;\;\;{\left(\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\ell + \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{elif}\;k\_m \leq 3.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{k\_m \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell + \ell}{t \cdot \left(k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.8e-97)
   (* (pow (/ (* (* k_m k_m) t) (+ l l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (if (<= k_m 3.4e+99)
     (*
      (/ (* l (cos k_m)) k_m)
      (/
       (* (/ l t) (fma 0.6666666666666666 (* k_m k_m) 2.0))
       (* k_m (* k_m k_m))))
     (*
      (/ l k_m)
      (/ (+ l l) (* t (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-97) {
		tmp = pow((((k_m * k_m) * t) / (l + l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else if (k_m <= 3.4e+99) {
		tmp = ((l * cos(k_m)) / k_m) * (((l / t) * fma(0.6666666666666666, (k_m * k_m), 2.0)) / (k_m * (k_m * k_m)));
	} else {
		tmp = (l / k_m) * ((l + l) / (t * (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-97)
		tmp = Float64((Float64(Float64(Float64(k_m * k_m) * t) / Float64(l + l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	elseif (k_m <= 3.4e+99)
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(Float64(l / t) * fma(0.6666666666666666, Float64(k_m * k_m), 2.0)) / Float64(k_m * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l + l) / Float64(t * Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.8e-97], N[(N[Power[N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(l + l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.4e+99], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.79999999999999999e-97

   1. Initial program 42.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      7. count-2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.0

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

   5. Simplified 65.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. unpow-prod-down N/A

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

      17. lower-*.f64 N/A

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

   7. Applied egg-rr 78.7%

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

   if 1.79999999999999999e-97 < k < 3.39999999999999984e99

   1. Initial program 22.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 83.0

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

   5. Simplified 83.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.7

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

   7. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. sqr-sin-a N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 94.7

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

   9. Applied egg-rr 94.7%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. cube-mult N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 N/A

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

   12. Simplified 78.1%

       \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)}} \]

   if 3.39999999999999984e99 < k

   1. Initial program 41.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.8

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

   5. Simplified 72.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 91.6

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

   7. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.5

         \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot t} \]

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 91.5

          \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), \color{blue}{-0.5}, 0.5\right)\right) \cdot t} \]

   9. Applied egg-rr 91.5%

      \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t}} \]

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 61.6

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t} \]

   12. Simplified 61.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-97}:\\ \;\;\;\;{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell + \ell}\right)}^{-1} \cdot {\left(\frac{k \cdot k}{\ell}\right)}^{-1}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell + \ell}{t \cdot \left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell + \ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 3.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{k\_m \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell + \ell}{t \cdot \left(k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.2e-102)
   (* (/ l (* k_m k_m)) (/ (+ l l) (* (* k_m k_m) t)))
   (if (<= k_m 3.4e+99)
     (*
      (/ (* l (cos k_m)) k_m)
      (/
       (* (/ l t) (fma 0.6666666666666666 (* k_m k_m) 2.0))
       (* k_m (* k_m k_m))))
     (*
      (/ l k_m)
      (/ (+ l l) (* t (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.2e-102) {
		tmp = (l / (k_m * k_m)) * ((l + l) / ((k_m * k_m) * t));
	} else if (k_m <= 3.4e+99) {
		tmp = ((l * cos(k_m)) / k_m) * (((l / t) * fma(0.6666666666666666, (k_m * k_m), 2.0)) / (k_m * (k_m * k_m)));
	} else {
		tmp = (l / k_m) * ((l + l) / (t * (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.2e-102)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l + l) / Float64(Float64(k_m * k_m) * t)));
	elseif (k_m <= 3.4e+99)
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(Float64(l / t) * fma(0.6666666666666666, Float64(k_m * k_m), 2.0)) / Float64(k_m * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l + l) / Float64(t * Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.2e-102], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.4e+99], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.2000000000000005e-102

   1. Initial program 42.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      7. count-2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.0

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

   5. Simplified 65.0%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 78.6

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

   7. Applied egg-rr 78.6%

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

   if 8.2000000000000005e-102 < k < 3.39999999999999984e99

   1. Initial program 22.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 83.0

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

   5. Simplified 83.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.7

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

   7. Applied egg-rr 75.8%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. sqr-sin-a N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 94.7

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

   9. Applied egg-rr 94.7%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. cube-mult N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 N/A

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

   12. Simplified 78.1%

       \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)}} \]

   if 3.39999999999999984e99 < k

   1. Initial program 41.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 72.8

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

   5. Simplified 72.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 91.6

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

   7. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.5

         \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot t} \]

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 91.5

          \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), \color{blue}{-0.5}, 0.5\right)\right) \cdot t} \]

   9. Applied egg-rr 91.5%

      \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t}} \]

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 61.6

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t} \]

   12. Simplified 61.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot t}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell + \ell}{t \cdot \left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+47}:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k\_m} \cdot \frac{\ell + \ell}{t \cdot \left(k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.95e+47)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (* (/ l k_m) (/ (+ l l) (* t (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e+47) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = (l / k_m) * ((l + l) / (t * (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e+47)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l / k_m) * Float64(Float64(l + l) / Float64(t * Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.95e+47], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.95000000000000013e47

   1. Initial program 40.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 52.8%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 1.95000000000000013e47 < k

   1. Initial program 33.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 71.2

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

   5. Simplified 71.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.8

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

   7. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.8

         \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)\right)} \cdot t} \]

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 90.8

          \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), \color{blue}{-0.5}, 0.5\right)\right) \cdot t} \]

   9. Applied egg-rr 90.8%

      \[\leadsto \frac{\cos k \cdot \ell}{k} \cdot \frac{\ell + \ell}{\color{blue}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t}} \]

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 58.0

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell + \ell}{\left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right) \cdot t} \]

   12. Simplified 58.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+47}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell + \ell}{t \cdot \left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.68:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m \cdot \left(\ell \cdot \left(\ell + \ell\right)\right)}{k\_m \cdot \left(k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.68)
   (*
    (/ l (* k_m k_m))
    (/
     (*
      l
      (fma
       k_m
       (*
        (* k_m (* k_m k_m))
        (fma k_m (* (/ k_m t) -0.0205026455026455) (/ -0.11666666666666667 t)))
       (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
     (* k_m k_m)))
   (/ (* (cos k_m) (* l (+ l l))) (* k_m (* k_m (* (* k_m k_m) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.68) {
		tmp = (l / (k_m * k_m)) * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = (cos(k_m) * (l * (l + l))) / (k_m * (k_m * ((k_m * k_m) * t)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.68)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(cos(k_m) * Float64(l * Float64(l + l))) / Float64(k_m * Float64(k_m * Float64(Float64(k_m * k_m) * t))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.68], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.680000000000000049

   1. Initial program 41.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 53.3%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 0.680000000000000049 < k

   1. Initial program 31.9%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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. lower-/.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-sin.f64 71.9

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

   5. Simplified 71.9%

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

   6. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.7

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

   8. Simplified 54.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.68:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \left(\ell \cdot \left(\ell + \ell\right)\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+47}:\\ \;\;\;\;t\_1 \cdot \frac{\ell \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \mathsf{fma}\left(k\_m, \frac{k\_m}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k\_m \cdot k\_m}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\ell + \ell}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= k_m 1.95e+47)
     (*
      t_1
      (/
       (*
        l
        (fma
         k_m
         (*
          (* k_m (* k_m k_m))
          (fma
           k_m
           (* (/ k_m t) -0.0205026455026455)
           (/ -0.11666666666666667 t)))
         (fma (/ (* k_m k_m) t) -0.3333333333333333 (/ 2.0 t))))
       (* k_m k_m)))
     (* t_1 (/ (+ l l) (* (* k_m k_m) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (k_m <= 1.95e+47) {
		tmp = t_1 * ((l * fma(k_m, ((k_m * (k_m * k_m)) * fma(k_m, ((k_m / t) * -0.0205026455026455), (-0.11666666666666667 / t))), fma(((k_m * k_m) / t), -0.3333333333333333, (2.0 / t)))) / (k_m * k_m));
	} else {
		tmp = t_1 * ((l + l) / ((k_m * k_m) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 1.95e+47)
		tmp = Float64(t_1 * Float64(Float64(l * fma(k_m, Float64(Float64(k_m * Float64(k_m * k_m)) * fma(k_m, Float64(Float64(k_m / t) * -0.0205026455026455), Float64(-0.11666666666666667 / t))), fma(Float64(Float64(k_m * k_m) / t), -0.3333333333333333, Float64(2.0 / t)))) / Float64(k_m * k_m)));
	else
		tmp = Float64(t_1 * Float64(Float64(l + l) / Float64(Float64(k_m * 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[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.95e+47], N[(t$95$1 * N[(N[(l * N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m / t), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] + N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333 + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l + l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.95000000000000013e47

   1. Initial program 40.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + \left(2 \cdot \frac{1}{t} + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right) + {k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left({k}^{4} \cdot \left(\frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}\right) + \left(\frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\mathsf{fma}\left({k}^{4}, \frac{-31}{1512} \cdot \frac{{k}^{2}}{t} - \frac{7}{60} \cdot \frac{1}{t}, \frac{-1}{3} \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   8. Simplified 52.8%

      \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(-0.0205026455026455, \frac{k \cdot k}{t}, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(-0.3333333333333333, \frac{k \cdot k}{t}, \frac{2}{t}\right)\right)}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \]

   10. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}} \]

   if 1.95000000000000013e47 < k

   1. Initial program 33.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      7. count-2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.4

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

   5. Simplified 52.4%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 55.4

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

   7. Applied egg-rr 55.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+47}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell \cdot \mathsf{fma}\left(k, \left(k \cdot \left(k \cdot k\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t} \cdot -0.0205026455026455, \frac{-0.11666666666666667}{t}\right), \mathsf{fma}\left(\frac{k \cdot k}{t}, -0.3333333333333333, \frac{2}{t}\right)\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m + k\_m}{t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{if}\;k\_m \leq 6 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;k\_m \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(k\_m \cdot -0.0205026455026455\right) \cdot \left(k\_m \cdot \left(\ell \cdot \ell\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (+ k_m k_m) (* t (* (* k_m k_m) (* k_m k_m))))))
   (if (<= k_m 6e-39)
     t_1
     (if (<= k_m 1.55e+81)
       (/ (* (* k_m -0.0205026455026455) (* k_m (* l l))) t)
       t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m + k_m) / (t * ((k_m * k_m) * (k_m * k_m)));
	double tmp;
	if (k_m <= 6e-39) {
		tmp = t_1;
	} else if (k_m <= 1.55e+81) {
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t;
	} else {
		tmp = 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 = (k_m + k_m) / (t * ((k_m * k_m) * (k_m * k_m)))
    if (k_m <= 6d-39) then
        tmp = t_1
    else if (k_m <= 1.55d+81) then
        tmp = ((k_m * (-0.0205026455026455d0)) * (k_m * (l * l))) / t
    else
        tmp = 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 = (k_m + k_m) / (t * ((k_m * k_m) * (k_m * k_m)));
	double tmp;
	if (k_m <= 6e-39) {
		tmp = t_1;
	} else if (k_m <= 1.55e+81) {
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m + k_m) / (t * ((k_m * k_m) * (k_m * k_m)))
	tmp = 0
	if k_m <= 6e-39:
		tmp = t_1
	elif k_m <= 1.55e+81:
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t
	else:
		tmp = t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m + k_m) / Float64(t * Float64(Float64(k_m * k_m) * Float64(k_m * k_m))))
	tmp = 0.0
	if (k_m <= 6e-39)
		tmp = t_1;
	elseif (k_m <= 1.55e+81)
		tmp = Float64(Float64(Float64(k_m * -0.0205026455026455) * Float64(k_m * Float64(l * l))) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m + k_m) / (t * ((k_m * k_m) * (k_m * k_m)));
	tmp = 0.0;
	if (k_m <= 6e-39)
		tmp = t_1;
	elseif (k_m <= 1.55e+81)
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t;
	else
		tmp = 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[(N[(k$95$m + k$95$m), $MachinePrecision] / N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 6e-39], t$95$1, If[LessEqual[k$95$m, 1.55e+81], N[(N[(N[(k$95$m * -0.0205026455026455), $MachinePrecision] * N[(k$95$m * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.00000000000000055e-39 or 1.55e81 < k

   1. Initial program 42.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      7. count-2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 64.5

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

   5. Simplified 64.5%

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

      1. flip-+ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      7. +-inverses N/A

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\ell - \ell\right)}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      8. distribute-lft-out-- N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      9. +-inverses N/A

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

      10. clear-num N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{\frac{1}{\frac{\color{blue}{0}}{\ell \cdot \ell - \ell \cdot \ell}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      12. +-inverses N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-inverses N/A

          \[\leadsto \frac{\frac{1}{\frac{\ell \cdot \ell - \ell \cdot \ell}{\color{blue}{0}}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. lift-+.f64 N/A

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

      21. lower-/.f64 40.9

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

   7. Applied egg-rr 40.9%

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

      1. flip-+ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\frac{1}{\frac{\color{blue}{0}}{\ell - \ell}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. +-inverses N/A

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

      6. +-inverses N/A

         \[\leadsto \frac{\frac{1}{\frac{k - k}{\color{blue}{0}}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      7. +-inverses N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. clear-num N/A

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

      11. flip-+ N/A

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

      12. lift-+.f64 41.4

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

   9. Applied egg-rr 41.4%

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

   if 6.00000000000000055e-39 < k < 1.55e81

   1. Initial program 7.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{\ell}^{2}}{t}\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \color{blue}{\left(k \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{{\ell}^{2}}{t}}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

      9. lower-*.f64 34.0

         \[\leadsto -0.0205026455026455 \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{-0.0205026455026455 \cdot \left(k \cdot \left(k \cdot \frac{\ell \cdot \ell}{t}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{\ell \cdot \ell}{t}}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \color{blue}{\left(k \cdot \frac{\ell \cdot \ell}{t}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \frac{\ell \cdot \ell}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-31}{1512} \cdot k\right) \cdot \color{blue}{\left(k \cdot \frac{\ell \cdot \ell}{t}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \color{blue}{\frac{\ell \cdot \ell}{t}}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\frac{-31}{1512} \cdot k\right) \cdot \color{blue}{\frac{k \cdot \left(\ell \cdot \ell\right)}{t}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}}{t} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(k \cdot \frac{-31}{1512}\right)} \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(k \cdot \frac{-31}{1512}\right)} \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t} \]

      13. lower-*.f64 34.0

          \[\leadsto \frac{\left(k \cdot -0.0205026455026455\right) \cdot \color{blue}{\left(k \cdot \left(\ell \cdot \ell\right)\right)}}{t} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\frac{\left(k \cdot -0.0205026455026455\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m + k\_m}{k\_m \cdot \left(\left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t\right)}\\ \mathbf{if}\;k\_m \leq 6 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;k\_m \leq 1.85 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(k\_m \cdot -0.0205026455026455\right) \cdot \left(k\_m \cdot \left(\ell \cdot \ell\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (+ k_m k_m) (* k_m (* (* k_m (* k_m k_m)) t)))))
   (if (<= k_m 6e-39)
     t_1
     (if (<= k_m 1.85e+81)
       (/ (* (* k_m -0.0205026455026455) (* k_m (* l l))) t)
       t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m + k_m) / (k_m * ((k_m * (k_m * k_m)) * t));
	double tmp;
	if (k_m <= 6e-39) {
		tmp = t_1;
	} else if (k_m <= 1.85e+81) {
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t;
	} else {
		tmp = 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 = (k_m + k_m) / (k_m * ((k_m * (k_m * k_m)) * t))
    if (k_m <= 6d-39) then
        tmp = t_1
    else if (k_m <= 1.85d+81) then
        tmp = ((k_m * (-0.0205026455026455d0)) * (k_m * (l * l))) / t
    else
        tmp = 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 = (k_m + k_m) / (k_m * ((k_m * (k_m * k_m)) * t));
	double tmp;
	if (k_m <= 6e-39) {
		tmp = t_1;
	} else if (k_m <= 1.85e+81) {
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m + k_m) / (k_m * ((k_m * (k_m * k_m)) * t))
	tmp = 0
	if k_m <= 6e-39:
		tmp = t_1
	elif k_m <= 1.85e+81:
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t
	else:
		tmp = t_1
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m + k_m) / Float64(k_m * Float64(Float64(k_m * Float64(k_m * k_m)) * t)))
	tmp = 0.0
	if (k_m <= 6e-39)
		tmp = t_1;
	elseif (k_m <= 1.85e+81)
		tmp = Float64(Float64(Float64(k_m * -0.0205026455026455) * Float64(k_m * Float64(l * l))) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m + k_m) / (k_m * ((k_m * (k_m * k_m)) * t));
	tmp = 0.0;
	if (k_m <= 6e-39)
		tmp = t_1;
	elseif (k_m <= 1.85e+81)
		tmp = ((k_m * -0.0205026455026455) * (k_m * (l * l))) / t;
	else
		tmp = 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[(N[(k$95$m + k$95$m), $MachinePrecision] / N[(k$95$m * N[(N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 6e-39], t$95$1, If[LessEqual[k$95$m, 1.85e+81], N[(N[(N[(k$95$m * -0.0205026455026455), $MachinePrecision] * N[(k$95$m * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.00000000000000055e-39 or 1.85e81 < k

   1. Initial program 42.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      7. count-2 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

      9. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 64.5

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

   5. Simplified 64.5%

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

      1. flip-+ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. +-inverses N/A

         \[\leadsto \frac{\ell \cdot \frac{0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      7. +-inverses N/A

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\ell - \ell\right)}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      8. distribute-lft-out-- N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      9. +-inverses N/A

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

      10. clear-num N/A

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

      11. +-inverses N/A

          \[\leadsto \frac{\frac{1}{\frac{\color{blue}{0}}{\ell \cdot \ell - \ell \cdot \ell}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      12. +-inverses N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-inverses N/A

          \[\leadsto \frac{\frac{1}{\frac{\ell \cdot \ell - \ell \cdot \ell}{\color{blue}{0}}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. lift-+.f64 N/A

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

      21. lower-/.f64 40.9

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

   7. Applied egg-rr 40.9%

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

      1. lift-+.f64 N/A

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

      2. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{\mathsf{neg}\left(\left(\ell + \ell\right)\right)}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 40.9

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

   9. Applied egg-rr 41.4%

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

   if 6.00000000000000055e-39 < k < 1.85e81

   1. Initial program 7.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{\ell}^{2}}{t}\right) \]

      4. associate-*l* N/A

         \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \color{blue}{\left(k \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{{\ell}^{2}}{t}}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

      9. lower-*.f64 34.0

         \[\leadsto -0.0205026455026455 \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   8. Simplified 34.0%

      \[\leadsto \color{blue}{-0.0205026455026455 \cdot \left(k \cdot \left(k \cdot \frac{\ell \cdot \ell}{t}\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{\ell \cdot \ell}{t}}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \color{blue}{\left(k \cdot \frac{\ell \cdot \ell}{t}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \frac{\ell \cdot \ell}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{-31}{1512} \cdot k\right) \cdot \color{blue}{\left(k \cdot \frac{\ell \cdot \ell}{t}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \color{blue}{\frac{\ell \cdot \ell}{t}}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\frac{-31}{1512} \cdot k\right) \cdot \color{blue}{\frac{k \cdot \left(\ell \cdot \ell\right)}{t}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-31}{1512} \cdot k\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}}{t} \]

      11. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(k \cdot \frac{-31}{1512}\right)} \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(k \cdot \frac{-31}{1512}\right)} \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t} \]

      13. lower-*.f64 34.0

          \[\leadsto \frac{\left(k \cdot -0.0205026455026455\right) \cdot \color{blue}{\left(k \cdot \left(\ell \cdot \ell\right)\right)}}{t} \]

   10. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\frac{\left(k \cdot -0.0205026455026455\right) \cdot \left(k \cdot \left(\ell \cdot \ell\right)\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{k\_m \cdot k\_m} \cdot \frac{\ell + \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m k_m)) (/ (+ l l) (* (* k_m k_m) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (k_m * k_m)) * ((l + l) / ((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 / (k_m * k_m)) * ((l + l) / ((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 / (k_m * k_m)) * ((l + l) / ((k_m * k_m) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (k_m * k_m)) * ((l + l) / ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(l + l) / Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / (k_m * k_m)) * ((l + l) / ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l + l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   7. count-2 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.3

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

5. Simplified 63.3%

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

   1. lift-+.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 74.1

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

7. Applied egg-rr 74.1%

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

8. Final simplification 74.1%

   \[\leadsto \frac{\ell}{k \cdot k} \cdot \frac{\ell + \ell}{\left(k \cdot k\right) \cdot t} \]
9. Add Preprocessing

Alternative 14: 68.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell + \ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ (+ l l) (* t (* k_m (* k_m (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * ((l + l) / (t * (k_m * (k_m * (k_m * k_m)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * ((l + l) / (t * (k_m * (k_m * (k_m * k_m)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * ((l + l) / (t * (k_m * (k_m * (k_m * k_m)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * ((l + l) / (t * (k_m * (k_m * (k_m * k_m)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(Float64(l + l) / Float64(t * Float64(k_m * Float64(k_m * Float64(k_m * k_m))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * ((l + l) / (t * (k_m * (k_m * (k_m * k_m)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   7. count-2 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.3

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

5. Simplified 63.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

7. Applied egg-rr 67.9%

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

8. Final simplification 67.9%

   \[\leadsto \ell \cdot \frac{\ell + \ell}{t \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)} \]
9. Add Preprocessing

Alternative 15: 43.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (/ l (* (* k_m k_m) (* k_m (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * (l / ((k_m * k_m) * (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 = 2.0d0 * (l / ((k_m * k_m) * (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 2.0 * (l / ((k_m * k_m) * (k_m * (k_m * t))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * (l / ((k_m * k_m) * (k_m * (k_m * t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64(l / Float64(Float64(k_m * k_m) * Float64(k_m * Float64(k_m * t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * (l / ((k_m * k_m) * (k_m * (k_m * t))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   7. count-2 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.3

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

5. Simplified 63.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. flip-+ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{\ell \cdot \frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   10. +-inverses N/A

       \[\leadsto \frac{\ell \cdot \frac{0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   12. +-inverses N/A

       \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\ell - \ell\right)}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   13. distribute-lft-out-- N/A

       \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   14. +-inverses N/A

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

   15. flip-+ N/A

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

   16. lift-+.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*r* N/A

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

   19. associate-/r* N/A

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

   20. lower-/.f64 N/A

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

7. Applied egg-rr 41.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-+.f64 N/A

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

   7. count-2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 41.0

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. lift-*.f64 N/A

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

   17. lower-*.f64 41.0

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

9. Applied egg-rr 41.0%

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

Alternative 16: 43.3% accurate, 13.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell + \ell}{t \cdot \left(k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (+ l l) (* t (* k_m (* k_m (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l + l) / (t * (k_m * (k_m * (k_m * k_m))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l + l) / (t * (k_m * (k_m * (k_m * k_m))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l + l) / (t * (k_m * (k_m * (k_m * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l + l) / (t * (k_m * (k_m * (k_m * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l + l) / Float64(t * Float64(k_m * Float64(k_m * Float64(k_m * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l + l) / (t * (k_m * (k_m * (k_m * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l + l), $MachinePrecision] / N[(t * N[(k$95$m * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{4} \cdot t} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \left(2 \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   7. count-2 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\left(\ell + \ell\right)}}{{k}^{4} \cdot t} \]

   9. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \left(\ell + \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 63.3

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

5. Simplified 63.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. flip-+ N/A

      \[\leadsto \frac{\ell \cdot \color{blue}{\frac{\ell \cdot \ell - \ell \cdot \ell}{\ell - \ell}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \frac{\color{blue}{\ell \cdot \ell} - \ell \cdot \ell}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \frac{\ell \cdot \ell - \color{blue}{\ell \cdot \ell}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   10. +-inverses N/A

       \[\leadsto \frac{\ell \cdot \frac{\color{blue}{0}}{\ell - \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   11. +-inverses N/A

       \[\leadsto \frac{\ell \cdot \frac{0}{\color{blue}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   12. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\ell \cdot 0}{0}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   13. +-inverses N/A

       \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\ell - \ell\right)}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   14. distribute-lft-out-- N/A

       \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell - \ell \cdot \ell}}{0}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   15. +-inverses N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell - \ell \cdot \ell}{\color{blue}{\ell - \ell}}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   16. flip-+ N/A

       \[\leadsto \frac{\color{blue}{\ell + \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\color{blue}{\ell + \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   18. lower-/.f64 41.0

       \[\leadsto \color{blue}{\frac{\ell + \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{\ell + \ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{\ell + \ell}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)} \]

7. Applied egg-rr 41.0%

   \[\leadsto \color{blue}{\frac{\ell + \ell}{t \cdot \left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}} \]
8. Add Preprocessing

Alternative 17: 20.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.0205026455026455 \cdot \left(k\_m \cdot \left(k\_m \cdot \frac{\ell \cdot \ell}{t}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* -0.0205026455026455 (* k_m (* k_m (/ (* l l) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.0205026455026455 * (k_m * (k_m * ((l * l) / 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 = (-0.0205026455026455d0) * (k_m * (k_m * ((l * l) / t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.0205026455026455 * (k_m * (k_m * ((l * l) / t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.0205026455026455 * (k_m * (k_m * ((l * l) / t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.0205026455026455 * Float64(k_m * Float64(k_m * Float64(Float64(l * l) / t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.0205026455026455 * (k_m * (k_m * ((l * l) / t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.0205026455026455 * N[(k$95$m * N[(k$95$m * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{6} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right) + \left(\frac{-31}{2160} \cdot \frac{{\ell}^{2}}{t} + \frac{173}{5040} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right) + -2 \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right)\right)}{{k}^{4}}} \]

5. Simplified 28.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right), \mathsf{fma}\left(k \cdot k, -2 \cdot \left(\frac{\ell \cdot \ell}{t} \cdot 0.01025132275132275\right), \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right), \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{-31}{1512} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{{\ell}^{2}}{t}\right) \]

   4. associate-*l* N/A

      \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(k \cdot \left(k \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \color{blue}{\left(k \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \color{blue}{\frac{{\ell}^{2}}{t}}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{-31}{1512} \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   9. lower-*.f64 17.8

      \[\leadsto -0.0205026455026455 \cdot \left(k \cdot \left(k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

8. Simplified 17.8%

   \[\leadsto \color{blue}{-0.0205026455026455 \cdot \left(k \cdot \left(k \cdot \frac{\ell \cdot \ell}{t}\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))))