Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 96.2%

Time: 16.9s

Alternatives: 20

Speedup: 11.0×

• 40.2% 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: 35.7% 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: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \ell \cdot \cos k\_m\\ \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{t\_1}{t \cdot {\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\frac{t\_1}{k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* l (cos k_m))))
   (if (<= k_m 4.6e+22)
     (* (/ (* 2.0 l) (* k_m k_m)) (/ t_1 (* t (pow (sin k_m) 2.0))))
     (*
      (/ (* 2.0 l) k_m)
      (/ (/ t_1 (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5))) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l * cos(k_m);
	double tmp;
	if (k_m <= 4.6e+22) {
		tmp = ((2.0 * l) / (k_m * k_m)) * (t_1 / (t * pow(sin(k_m), 2.0)));
	} else {
		tmp = ((2.0 * l) / k_m) * ((t_1 / (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))) / t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l * cos(k_m))
	tmp = 0.0
	if (k_m <= 4.6e+22)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(k_m * k_m)) * Float64(t_1 / Float64(t * (sin(k_m) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(Float64(t_1 / Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5))) / 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[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 4.6e+22], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(t$95$1 / N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6000000000000004e22

   1. Initial program 36.4%

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 78.3

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

   5. Simplified 78.3%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. sqr-sin-a N/A

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

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

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

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

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

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

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

      19. +-lowering-+.f64 76.1

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

   7. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. sqr-sin-a N/A

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

      3. pow2 N/A

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

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

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

      5. sin-lowering-sin.f64 88.9

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

   9. Applied egg-rr 88.9%

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

   if 4.6000000000000004e22 < k

   1. Initial program 29.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 72.6

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

   5. Simplified 72.6%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. sqr-sin-a N/A

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

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

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

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

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

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

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

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

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

      20. *-lowering-*.f64 93.5

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

   7. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. sqr-sin-a N/A

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

      3. pow2 N/A

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

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

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

      5. sin-lowering-sin.f64 93.6

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

   9. Applied egg-rr 93.6%

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

       1. associate-*r* N/A

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

       2. associate-/r* N/A

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

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

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

   11. Applied egg-rr 99.4%

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

Alternative 2: 91.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos \left(k\_m + k\_m\right)\\ \mathbf{if}\;k\_m \leq 0.000118:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{elif}\;k\_m \leq 3.8 \cdot 10^{+113}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k\_m}{\left(0.5 - t\_1 \cdot 0.5\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2 \cdot \ell}{k\_m}\right) \cdot \frac{\cos k\_m}{k\_m \cdot \left(t \cdot \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (cos (+ k_m k_m))))
   (if (<= k_m 0.000118)
     (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
     (if (<= k_m 3.8e+113)
       (*
        (* 2.0 l)
        (/ (* l (cos k_m)) (* (- 0.5 (* t_1 0.5)) (* (* k_m k_m) t))))
       (*
        (* l (/ (* 2.0 l) k_m))
        (/ (cos k_m) (* k_m (* t (fma t_1 -0.5 0.5)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos((k_m + k_m));
	double tmp;
	if (k_m <= 0.000118) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else if (k_m <= 3.8e+113) {
		tmp = (2.0 * l) * ((l * cos(k_m)) / ((0.5 - (t_1 * 0.5)) * ((k_m * k_m) * t)));
	} else {
		tmp = (l * ((2.0 * l) / k_m)) * (cos(k_m) / (k_m * (t * fma(t_1, -0.5, 0.5))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cos(Float64(k_m + k_m))
	tmp = 0.0
	if (k_m <= 0.000118)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	elseif (k_m <= 3.8e+113)
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(l * cos(k_m)) / Float64(Float64(0.5 - Float64(t_1 * 0.5)) * Float64(Float64(k_m * k_m) * t))));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 * l) / k_m)) * Float64(cos(k_m) / Float64(k_m * Float64(t * fma(t_1, -0.5, 0.5)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k$95$m, 0.000118], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.8e+113], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(k$95$m * N[(t * N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.18e-4

   1. Initial program 36.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.9

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

   5. Simplified 67.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 1.18e-4 < k < 3.8000000000000003e113

   1. Initial program 8.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 75.7

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

   5. Simplified 75.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. sqr-sin-a N/A

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

      16. --lowering--.f64 N/A

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

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

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

      18. cos-2 N/A

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

      19. cos-sum N/A

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

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

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

      21. +-lowering-+.f64 99.4

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

   7. Applied egg-rr 99.4%

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

   if 3.8000000000000003e113 < k

   1. Initial program 39.7%

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 67.8

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

   5. Simplified 67.8%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. sqr-sin-a N/A

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

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

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

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

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

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

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

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

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

      20. *-lowering-*.f64 90.7

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

   7. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. sqr-sin-a N/A

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

      3. pow2 N/A

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

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

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

      5. sin-lowering-sin.f64 90.8

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

   9. Applied egg-rr 90.8%

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l* N/A

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

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

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

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

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

       13. unpow2 N/A

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

       14. sqr-sin-a N/A

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

       15. count-2 N/A

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

       16. sub-neg N/A

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

       17. +-commutative N/A

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

       18. *-commutative N/A

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

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

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

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

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

   11. Applied egg-rr 88.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000118:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+113}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2 \cdot \ell}{k}\right) \cdot \frac{\cos k}{k \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\frac{\ell \cdot \cos k\_m}{k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.8e-5)
   (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
   (*
    (/ (* 2.0 l) k_m)
    (/ (/ (* l (cos k_m)) (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5))) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.8e-5) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = ((2.0 * l) / k_m) * (((l * cos(k_m)) / (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))) / t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.8e-5)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(Float64(Float64(l * cos(k_m)) / Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5))) / t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.8e-5], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.7999999999999998e-5

   1. Initial program 36.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.9

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

   5. Simplified 67.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 8.7999999999999998e-5 < k

   1. Initial program 28.9%

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 70.6

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

   5. Simplified 70.6%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. sqr-sin-a N/A

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

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

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

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

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

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

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

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

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

      20. *-lowering-*.f64 93.7

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

   7. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. sqr-sin-a N/A

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

      3. pow2 N/A

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

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

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

      5. sin-lowering-sin.f64 93.8

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

   9. Applied egg-rr 93.8%

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

       1. associate-*r* N/A

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

       2. associate-/r* N/A

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

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

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

   11. Applied egg-rr 99.4%

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

Alternative 4: 93.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\ell \cdot \cos k\_m}{k\_m \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.8e-5)
   (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
   (*
    (/ (* 2.0 l) k_m)
    (/ (* l (cos k_m)) (* k_m (* t (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 <= 8.8e-5) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = ((2.0 * l) / k_m) * ((l * cos(k_m)) / (k_m * (t * 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 <= 8.8e-5)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(Float64(l * cos(k_m)) / Float64(k_m * Float64(t * 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, 8.8e-5], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(t * 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 < 8.7999999999999998e-5

   1. Initial program 36.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.9

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

   5. Simplified 67.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 8.7999999999999998e-5 < k

   1. Initial program 28.9%

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 70.6

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

   5. Simplified 70.6%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. sqr-sin-a N/A

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

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

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

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

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

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

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

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

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

      20. *-lowering-*.f64 93.7

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

   7. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. sqr-sin-a N/A

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

      3. pow2 N/A

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

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

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

      5. sin-lowering-sin.f64 93.8

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

   9. Applied egg-rr 93.8%

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

       1. *-commutative N/A

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

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

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

   11. Applied egg-rr 93.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\ell \cdot \cos k\_m}{t \cdot \left(k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8e-5)
   (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
   (*
    (/ (* 2.0 l) k_m)
    (/ (* l (cos k_m)) (* 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 <= 8e-5) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = ((2.0 * l) / k_m) * ((l * cos(k_m)) / (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 <= 8e-5)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(Float64(l * cos(k_m)) / 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, 8e-5], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $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 < 8.00000000000000065e-5

   1. Initial program 36.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.9

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

   5. Simplified 67.9%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 70.3

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

   7. Applied egg-rr 70.3%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 8.00000000000000065e-5 < k

   1. Initial program 28.9%

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

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. sin-lowering-sin.f64 70.6

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

   5. Simplified 70.6%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. sqr-sin-a N/A

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

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

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

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

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

      16. cos-2 N/A

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

      17. cos-sum N/A

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

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

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

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

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

      20. *-lowering-*.f64 93.7

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

   7. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

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

      2. sqr-sin-a N/A

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

      3. pow2 N/A

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

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

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

      5. sin-lowering-sin.f64 93.8

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

   9. Applied egg-rr 93.8%

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

       1. associate-*r* N/A

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

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

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

       6. sqr-sin-a N/A

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

       7. count-2 N/A

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

       8. sub-neg N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

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

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

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

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

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

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

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

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

       15. metadata-eval 93.7

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

   11. Applied egg-rr 93.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{elif}\;k\_m \leq 6.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot t}}{\sin k\_m \cdot \tan k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.5e-8)
   (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
   (if (<= k_m 6.5e+79)
     (/ (/ (* 2.0 (* l l)) (* (* k_m k_m) t)) (* (sin k_m) (tan k_m)))
     (*
      (/ (* 2.0 l) k_m)
      (/ l (* (* k_m t) (- 0.5 (* (cos (+ k_m k_m)) 0.5))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.5e-8) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else if (k_m <= 6.5e+79) {
		tmp = ((2.0 * (l * l)) / ((k_m * k_m) * t)) / (sin(k_m) * tan(k_m));
	} else {
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (cos((k_m + k_m)) * 0.5))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.5d-8) then
        tmp = 1.0d0 / ((((k_m * k_m) * 0.5d0) / l) * ((k_m * (k_m * t)) / l))
    else if (k_m <= 6.5d+79) then
        tmp = ((2.0d0 * (l * l)) / ((k_m * k_m) * t)) / (sin(k_m) * tan(k_m))
    else
        tmp = ((2.0d0 * l) / k_m) * (l / ((k_m * t) * (0.5d0 - (cos((k_m + k_m)) * 0.5d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.5e-8) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else if (k_m <= 6.5e+79) {
		tmp = ((2.0 * (l * l)) / ((k_m * k_m) * t)) / (Math.sin(k_m) * Math.tan(k_m));
	} else {
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (Math.cos((k_m + k_m)) * 0.5))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.5e-8:
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l))
	elif k_m <= 6.5e+79:
		tmp = ((2.0 * (l * l)) / ((k_m * k_m) * t)) / (math.sin(k_m) * math.tan(k_m))
	else:
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (math.cos((k_m + k_m)) * 0.5))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.5e-8)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	elseif (k_m <= 6.5e+79)
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(k_m * k_m) * t)) / Float64(sin(k_m) * tan(k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(l / Float64(Float64(k_m * t) * Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.5e-8)
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	elseif (k_m <= 6.5e+79)
		tmp = ((2.0 * (l * l)) / ((k_m * k_m) * t)) / (sin(k_m) * tan(k_m));
	else
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (cos((k_m + k_m)) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.5e-8], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6.5e+79], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.49999999999999993e-8

   1. Initial program 36.4%

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.8

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

   5. Simplified 67.8%

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

      1. clear-num N/A

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

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

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      11. *-lowering-*.f64 70.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      6. div-inv N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{2}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 4.49999999999999993e-8 < k < 6.49999999999999954e79

   1. Initial program 12.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \tan k}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\sin k \cdot \tan k}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \frac{1}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}}{\sin k \cdot \tan k}} \]

   4. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{\ell \cdot \ell}{t \cdot \left(t \cdot t\right)}\right) \cdot \frac{1}{\frac{k \cdot k}{t \cdot t}}}{\sin k \cdot \tan k}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}}{\sin k \cdot \tan k} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}}{\sin k \cdot \tan k} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}}{\sin k \cdot \tan k} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{2} \cdot t}}{\sin k \cdot \tan k} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot t}}{\sin k \cdot \tan k} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot t}}{\sin k \cdot \tan k} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{{k}^{2} \cdot t}}}{\sin k \cdot \tan k} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t}}{\sin k \cdot \tan k} \]

      8. *-lowering-*.f64 81.0

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t}}{\sin k \cdot \tan k} \]

   7. Simplified 81.0%

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t}}}{\sin k \cdot \tan k} \]

   if 6.49999999999999954e79 < k

   1. Initial program 35.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 68.1

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k}} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      16. cos-2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right) \cdot \left(k \cdot t\right)} \]

      17. cos-sum N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      18. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      20. *-lowering-*.f64 92.2

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \]
   9. Step-by-step derivation

   10. Simplified 67.4%

       \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k\_m \cdot \left(k\_m \cdot t\right)\\ \mathbf{if}\;k\_m \leq 0.000118:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{t\_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos k\_m\right) \cdot \left(\ell \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (* k_m t))))
   (if (<= k_m 0.000118)
     (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ t_1 l)))
     (*
      (* l (cos k_m))
      (* l (/ 2.0 (* (fma (cos (+ k_m k_m)) -0.5 0.5) t_1)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double tmp;
	if (k_m <= 0.000118) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * (t_1 / l));
	} else {
		tmp = (l * cos(k_m)) * (l * (2.0 / (fma(cos((k_m + k_m)), -0.5, 0.5) * t_1)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(k_m * t))
	tmp = 0.0
	if (k_m <= 0.000118)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(t_1 / l)));
	else
		tmp = Float64(Float64(l * cos(k_m)) * Float64(l * Float64(2.0 / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t_1))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 0.000118], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.18e-4

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 67.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      11. *-lowering-*.f64 70.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      6. div-inv N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{2}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 1.18e-4 < k

   1. Initial program 28.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 70.6

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k}} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      16. cos-2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right) \cdot \left(k \cdot t\right)} \]

      17. cos-sum N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      18. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      20. *-lowering-*.f64 93.7

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

      5. sin-lowering-sin.f64 93.8

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(k \cdot t\right)} \]

   9. Applied egg-rr 93.8%

      \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

   11. Applied egg-rr 82.4%

       \[\leadsto \color{blue}{\left(\frac{2}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000118:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos k\right) \cdot \left(\ell \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k\_m \cdot \left(k\_m \cdot t\right)\\ \mathbf{if}\;k\_m \leq 0.00045:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{t\_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k\_m \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (* k_m t))))
   (if (<= k_m 0.00045)
     (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ t_1 l)))
     (*
      (* l l)
      (* (cos k_m) (/ 2.0 (* (fma (cos (+ k_m k_m)) -0.5 0.5) t_1)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double tmp;
	if (k_m <= 0.00045) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * (t_1 / l));
	} else {
		tmp = (l * l) * (cos(k_m) * (2.0 / (fma(cos((k_m + k_m)), -0.5, 0.5) * t_1)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(k_m * t))
	tmp = 0.0
	if (k_m <= 0.00045)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(t_1 / l)));
	else
		tmp = Float64(Float64(l * l) * Float64(cos(k_m) * Float64(2.0 / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * t_1))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 0.00045], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4999999999999999e-4

   1. Initial program 36.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 67.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      11. *-lowering-*.f64 70.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      6. div-inv N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{2}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      14. *-lowering-*.f64 80.3

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 4.4999999999999999e-4 < k

   1. Initial program 28.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 70.6

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k}} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      16. cos-2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right) \cdot \left(k \cdot t\right)} \]

      17. cos-sum N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      18. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      20. *-lowering-*.f64 93.7

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

      5. sin-lowering-sin.f64 93.8

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(k \cdot t\right)} \]

   9. Applied egg-rr 93.8%

      \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

   11. Applied egg-rr 70.6%

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{+230}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\ell}{{\sin k\_m}^{2} \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k\_m}{k\_m}}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 3.1e+230)
   (* (/ (* 2.0 l) k_m) (/ l (* (pow (sin k_m) 2.0) (* k_m t))))
   (/ (/ (* (* 2.0 l) (/ (* l (cos k_m)) k_m)) (* k_m (* k_m t))) k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 3.1e+230) {
		tmp = ((2.0 * l) / k_m) * (l / (pow(sin(k_m), 2.0) * (k_m * t)));
	} else {
		tmp = (((2.0 * l) * ((l * cos(k_m)) / k_m)) / (k_m * (k_m * t))) / k_m;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 3.1d+230) then
        tmp = ((2.0d0 * l) / k_m) * (l / ((sin(k_m) ** 2.0d0) * (k_m * t)))
    else
        tmp = (((2.0d0 * l) * ((l * cos(k_m)) / k_m)) / (k_m * (k_m * t))) / k_m
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 3.1e+230) {
		tmp = ((2.0 * l) / k_m) * (l / (Math.pow(Math.sin(k_m), 2.0) * (k_m * t)));
	} else {
		tmp = (((2.0 * l) * ((l * Math.cos(k_m)) / k_m)) / (k_m * (k_m * t))) / k_m;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 3.1e+230:
		tmp = ((2.0 * l) / k_m) * (l / (math.pow(math.sin(k_m), 2.0) * (k_m * t)))
	else:
		tmp = (((2.0 * l) * ((l * math.cos(k_m)) / k_m)) / (k_m * (k_m * t))) / k_m
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 3.1e+230)
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(l / Float64((sin(k_m) ^ 2.0) * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * l) * Float64(Float64(l * cos(k_m)) / k_m)) / Float64(k_m * Float64(k_m * t))) / k_m);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 3.1e+230)
		tmp = ((2.0 * l) / k_m) * (l / ((sin(k_m) ^ 2.0) * (k_m * t)));
	else
		tmp = (((2.0 * l) * ((l * cos(k_m)) / k_m)) / (k_m * (k_m * t))) / k_m;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 3.1e+230], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.09999999999999981e230

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 76.7

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k}} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      16. cos-2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right) \cdot \left(k \cdot t\right)} \]

      17. cos-sum N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      18. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      20. *-lowering-*.f64 84.6

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]
   8. Step-by-step derivation

      1. count-2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

      5. sin-lowering-sin.f64 94.1

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{{\color{blue}{\sin k}}^{2} \cdot \left(k \cdot t\right)} \]

   9. Applied egg-rr 94.1%

      \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2}} \cdot \left(k \cdot t\right)} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{{\sin k}^{2} \cdot \left(k \cdot t\right)} \]
   11. Step-by-step derivation

   12. Simplified 78.1%

       \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{{\sin k}^{2} \cdot \left(k \cdot t\right)} \]

   if 3.09999999999999981e230 < l

   1. Initial program 42.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 78.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      3. *-lowering-*.f64 63.2

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   8. Simplified 63.2%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{k} \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2 \cdot \ell}{k} \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2 \cdot \ell}{k} \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right)} \cdot k} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k} \cdot \left(\ell \cdot \cos k\right)}{t \cdot \left(k \cdot k\right)}}{k}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{2 \cdot \ell}{k} \cdot \left(\ell \cdot \cos k\right)}{t \cdot \left(k \cdot k\right)}}{k}} \]

   10. Applied egg-rr 74.1%

       \[\leadsto \color{blue}{\frac{\frac{\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k}{k}}{k \cdot \left(k \cdot t\right)}}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25e+42)
   (* (/ (* 2.0 l) (* k_m k_m)) (/ (* l (cos k_m)) (* (* k_m k_m) t)))
   (*
    (/ (* 2.0 l) k_m)
    (/ l (* (* k_m t) (- 0.5 (* (cos (+ k_m k_m)) 0.5)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+42) {
		tmp = ((2.0 * l) / (k_m * k_m)) * ((l * cos(k_m)) / ((k_m * k_m) * t));
	} else {
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (cos((k_m + k_m)) * 0.5))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d+42) then
        tmp = ((2.0d0 * l) / (k_m * k_m)) * ((l * cos(k_m)) / ((k_m * k_m) * t))
    else
        tmp = ((2.0d0 * l) / k_m) * (l / ((k_m * t) * (0.5d0 - (cos((k_m + k_m)) * 0.5d0))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25e+42) {
		tmp = ((2.0 * l) / (k_m * k_m)) * ((l * Math.cos(k_m)) / ((k_m * k_m) * t));
	} else {
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (Math.cos((k_m + k_m)) * 0.5))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25e+42:
		tmp = ((2.0 * l) / (k_m * k_m)) * ((l * math.cos(k_m)) / ((k_m * k_m) * t))
	else:
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (math.cos((k_m + k_m)) * 0.5))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25e+42)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(k_m * k_m)) * Float64(Float64(l * cos(k_m)) / Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(l / Float64(Float64(k_m * t) * Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25e+42)
		tmp = ((2.0 * l) / (k_m * k_m)) * ((l * cos(k_m)) / ((k_m * k_m) * t));
	else
		tmp = ((2.0 * l) / k_m) * (l / ((k_m * t) * (0.5 - (cos((k_m + k_m)) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25e+42], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000002e42

   1. Initial program 35.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 79.0

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 79.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{t \cdot {\sin k}^{2}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)} \]

      16. cos-2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right)} \]

      17. cos-sum N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right)} \]

      18. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right)} \]

      19. +-lowering-+.f64 76.8

          \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(k + k\right)}\right)} \]

   7. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{k}^{2} \cdot t}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

      3. *-lowering-*.f64 81.2

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   10. Simplified 81.2%

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   if 1.25000000000000002e42 < 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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 69.7

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k}} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k} \cdot \frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot t\right) \cdot {\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{\sin k}^{2} \cdot \left(k \cdot t\right)}} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\sin k \cdot \sin k\right)} \cdot \left(k \cdot t\right)} \]

      13. sqr-sin-a N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      14. --lowering--.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)} \cdot \left(k \cdot t\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \]

      16. cos-2 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right) \cdot \left(k \cdot t\right)} \]

      17. cos-sum N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      18. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(k + k\right)}\right) \cdot \left(k \cdot t\right)} \]

      20. *-lowering-*.f64 92.9

          \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]

   7. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \]
   9. Step-by-step derivation

   10. Simplified 63.2%

       \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\color{blue}{\ell}}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot \left(k \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(0.5 - \cos \left(k + k\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k\_m \cdot \left(k\_m \cdot t\right)\\ \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{t\_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot \left(\ell \cdot \cos k\_m\right)\right)}{k\_m \cdot t\_1}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (* k_m t))))
   (if (<= l 4.5e+174)
     (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ t_1 l)))
     (/ (/ (* l (* 2.0 (* l (cos k_m)))) (* k_m t_1)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double tmp;
	if (l <= 4.5e+174) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * (t_1 / l));
	} else {
		tmp = ((l * (2.0 * (l * cos(k_m)))) / (k_m * t_1)) / k_m;
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k_m * (k_m * t)
    if (l <= 4.5d+174) then
        tmp = 1.0d0 / ((((k_m * k_m) * 0.5d0) / l) * (t_1 / l))
    else
        tmp = ((l * (2.0d0 * (l * cos(k_m)))) / (k_m * t_1)) / k_m
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = k_m * (k_m * t);
	double tmp;
	if (l <= 4.5e+174) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * (t_1 / l));
	} else {
		tmp = ((l * (2.0 * (l * Math.cos(k_m)))) / (k_m * t_1)) / k_m;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = k_m * (k_m * t)
	tmp = 0
	if l <= 4.5e+174:
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * (t_1 / l))
	else:
		tmp = ((l * (2.0 * (l * math.cos(k_m)))) / (k_m * t_1)) / k_m
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(k_m * t))
	tmp = 0.0
	if (l <= 4.5e+174)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(t_1 / l)));
	else
		tmp = Float64(Float64(Float64(l * Float64(2.0 * Float64(l * cos(k_m)))) / Float64(k_m * t_1)) / k_m);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = k_m * (k_m * t);
	tmp = 0.0;
	if (l <= 4.5e+174)
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * (t_1 / l));
	else
		tmp = ((l * (2.0 * (l * cos(k_m)))) / (k_m * t_1)) / k_m;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 4.5e+174], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(2.0 * N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * t$95$1), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.50000000000000042e174

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 66.7

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      11. *-lowering-*.f64 68.8

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      6. div-inv N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{2}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      14. *-lowering-*.f64 77.0

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 77.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 4.50000000000000042e174 < l

   1. Initial program 34.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 69.2

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      3. *-lowering-*.f64 56.3

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   8. Simplified 56.3%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right) \cdot k}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}}{k} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right) \cdot 2}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\left(\left(\ell \cdot \cos k\right) \cdot 2\right)}}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell \cdot \left(\color{blue}{\left(\ell \cdot \cos k\right)} \cdot 2\right)}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \color{blue}{\cos k}\right) \cdot 2\right)}{k \cdot \left(\left(k \cdot k\right) \cdot t\right)}}{k} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{k \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}}{k} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{\color{blue}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}}}{k} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}}}{k} \]

      14. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}}{k} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}}}{k} \]

      16. *-lowering-*.f64 66.0

          \[\leadsto \frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{k \cdot \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)}}{k} \]

   10. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\left(\ell \cdot \cos k\right) \cdot 2\right)}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+174}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\_m\right)\right)}{k\_m \cdot \left(k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 6e+174)
   (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
   (/ (* 2.0 (* l (* l (cos k_m)))) (* 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 (l <= 6e+174) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = (2.0 * (l * (l * cos(k_m)))) / (k_m * (k_m * ((k_m * k_m) * t)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 6d+174) then
        tmp = 1.0d0 / ((((k_m * k_m) * 0.5d0) / l) * ((k_m * (k_m * t)) / l))
    else
        tmp = (2.0d0 * (l * (l * cos(k_m)))) / (k_m * (k_m * ((k_m * k_m) * t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 6e+174) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = (2.0 * (l * (l * Math.cos(k_m)))) / (k_m * (k_m * ((k_m * k_m) * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 6e+174:
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l))
	else:
		tmp = (2.0 * (l * (l * math.cos(k_m)))) / (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 (l <= 6e+174)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	else
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k_m)))) / Float64(k_m * Float64(k_m * Float64(Float64(k_m * k_m) * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 6e+174)
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	else
		tmp = (2.0 * (l * (l * cos(k_m)))) / (k_m * (k_m * ((k_m * k_m) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 6e+174], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * N[(l * N[Cos[k$95$m], $MachinePrecision]), $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 l < 6e174

   1. Initial program 34.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 66.7

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      11. *-lowering-*.f64 68.8

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      6. div-inv N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{2}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      14. *-lowering-*.f64 77.0

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 77.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 6e174 < l

   1. Initial program 34.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 69.2

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      3. *-lowering-*.f64 56.3

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   8. Simplified 56.3%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\ell \cdot \cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* 2.0 l) (* k_m k_m)) (/ (* l (cos 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)) * ((l * cos(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)) * ((l * cos(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)) * ((l * Math.cos(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)) * ((l * math.cos(k_m)) / ((k_m * k_m) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(2.0 * l) / Float64(k_m * k_m)) * Float64(Float64(l * cos(k_m)) / Float64(Float64(k_m * k_m) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 * l) / (k_m * k_m)) * ((l * cos(k_m)) / ((k_m * k_m) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   12. associate-*l* N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

   15. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

   18. pow-lowering-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

   19. sin-lowering-sin.f64 76.9

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

5. Simplified 76.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \cos k}}{t \cdot {\sin k}^{2}} \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \color{blue}{\cos k}}{t \cdot {\sin k}^{2}} \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{t \cdot {\sin k}^{2}}} \]

   12. unpow2 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}} \]

   13. sqr-sin-a N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}} \]

   14. --lowering--.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot k\right)\right)}} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot k\right)}\right)} \]

   16. cos-2 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)}\right)} \]

   17. cos-sum N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right)} \]

   18. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(k + k\right)}\right)} \]

   19. +-lowering-+.f64 76.7

       \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(k + k\right)}\right)} \]

7. Applied egg-rr 76.7%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{t \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{{k}^{2} \cdot t}} \]

   2. unpow2 N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   3. *-lowering-*.f64 75.4

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

10. Simplified 75.4%

    \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
11. Add Preprocessing

Alternative 14: 73.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{+230}:\\ \;\;\;\;\frac{1}{\frac{\left(k\_m \cdot k\_m\right) \cdot 0.5}{\ell} \cdot \frac{k\_m \cdot \left(k\_m \cdot t\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 - k\_m \cdot k\_m\right)}{k\_m \cdot \left(k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1.5e+230)
   (/ 1.0 (* (/ (* (* k_m k_m) 0.5) l) (/ (* k_m (* k_m t)) l)))
   (/ (* (* l l) (- 2.0 (* k_m k_m))) (* 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 (l <= 1.5e+230) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * ((k_m * k_m) * t)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 1.5d+230) then
        tmp = 1.0d0 / ((((k_m * k_m) * 0.5d0) / l) * ((k_m * (k_m * t)) / l))
    else
        tmp = ((l * l) * (2.0d0 - (k_m * k_m))) / (k_m * (k_m * ((k_m * k_m) * t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1.5e+230) {
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	} else {
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * ((k_m * k_m) * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1.5e+230:
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l))
	else:
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (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 (l <= 1.5e+230)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(k_m * k_m) * 0.5) / l) * Float64(Float64(k_m * Float64(k_m * t)) / l)));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 - Float64(k_m * k_m))) / Float64(k_m * Float64(k_m * Float64(Float64(k_m * k_m) * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1.5e+230)
		tmp = 1.0 / ((((k_m * k_m) * 0.5) / l) * ((k_m * (k_m * t)) / l));
	else
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * ((k_m * k_m) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1.5e+230], N[(1.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 - N[(k$95$m * k$95$m), $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 l < 1.50000000000000004e230

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 64.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)}{2 \cdot \left(\ell \cdot \ell\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}} \]

      11. *-lowering-*.f64 67.6

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{2 \cdot \left(\ell \cdot \ell\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{k \cdot k}{2 \cdot \ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      4. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{k \cdot k}{2}}{\ell}} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      6. div-inv N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right) \cdot \frac{1}{2}}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \frac{1}{2}}{\ell} \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot \frac{1}{2}}{\ell} \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell}} \]

      14. *-lowering-*.f64 75.0

          \[\leadsto \frac{1}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell}} \]

   9. Applied egg-rr 75.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(k \cdot k\right) \cdot 0.5}{\ell} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

   if 1.50000000000000004e230 < l

   1. Initial program 42.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 78.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      3. *-lowering-*.f64 63.2

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   8. Simplified 63.2%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2} + -1 \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \color{blue}{\left(-1 \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(2 + -1 \cdot {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(2 + -1 \cdot {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 + -1 \cdot {k}^{2}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 + -1 \cdot {k}^{2}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left({k}^{2}\right)\right)}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       8. unsub-neg N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 - {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       9. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 - {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{k \cdot k}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       11. *-lowering-*.f64 63.2

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{k \cdot k}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   11. Simplified 63.2%

       \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \left(2 - k \cdot k\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 73.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \left(k\_m \cdot k\_m\right) \cdot t\\ \mathbf{if}\;\ell \leq 3 \cdot 10^{+230}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_1} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 - k\_m \cdot k\_m\right)}{k\_m \cdot \left(k\_m \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (* k_m k_m) t)))
   (if (<= l 3e+230)
     (* (/ (* 2.0 l) t_1) (/ l (* k_m k_m)))
     (/ (* (* l l) (- 2.0 (* k_m k_m))) (* k_m (* k_m t_1))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double tmp;
	if (l <= 3e+230) {
		tmp = ((2.0 * l) / t_1) * (l / (k_m * k_m));
	} else {
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * 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
    if (l <= 3d+230) then
        tmp = ((2.0d0 * l) / t_1) * (l / (k_m * k_m))
    else
        tmp = ((l * l) * (2.0d0 - (k_m * k_m))) / (k_m * (k_m * 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;
	double tmp;
	if (l <= 3e+230) {
		tmp = ((2.0 * l) / t_1) * (l / (k_m * k_m));
	} else {
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * t_1));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m * k_m) * t
	tmp = 0
	if l <= 3e+230:
		tmp = ((2.0 * l) / t_1) * (l / (k_m * k_m))
	else:
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * t_1))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m * k_m) * t)
	tmp = 0.0
	if (l <= 3e+230)
		tmp = Float64(Float64(Float64(2.0 * l) / t_1) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 - Float64(k_m * k_m))) / Float64(k_m * Float64(k_m * 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;
	tmp = 0.0;
	if (l <= 3e+230)
		tmp = ((2.0 * l) / t_1) * (l / (k_m * k_m));
	else
		tmp = ((l * l) * (2.0 - (k_m * k_m))) / (k_m * (k_m * 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] * t), $MachinePrecision]}, If[LessEqual[l, 3e+230], N[(N[(N[(2.0 * l), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 - N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 3.00000000000000008e230

   1. Initial program 34.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 64.8

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{t \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      10. *-lowering-*.f64 74.9

          \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   7. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

   if 3.00000000000000008e230 < l

   1. Initial program 42.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\cos k \cdot \ell\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \left(\cos k \cdot \ell\right)\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

      18. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

      19. sin-lowering-sin.f64 78.9

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

      3. *-lowering-*.f64 63.2

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]

   8. Simplified 63.2%

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 2 \cdot {\ell}^{2}}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2} + -1 \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2} + \color{blue}{\left(-1 \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(2 + -1 \cdot {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \left(2 + -1 \cdot {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 + -1 \cdot {k}^{2}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 + -1 \cdot {k}^{2}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       7. mul-1-neg N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left({k}^{2}\right)\right)}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       8. unsub-neg N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 - {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       9. --lowering--.f64 N/A

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 - {k}^{2}\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{k \cdot k}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

       11. *-lowering-*.f64 63.2

           \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \left(2 - \color{blue}{k \cdot k}\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]

   11. Simplified 63.2%

       \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \left(2 - k \cdot k\right)}}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+230}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \left(2 - k \cdot k\right)}{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.1e+50)
   (* (/ (* 2.0 l) (* (* k_m k_m) t)) (/ l (* k_m k_m)))
   (/ (/ (* 2.0 (* l l)) (* k_m (* k_m t))) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+50) {
		tmp = ((2.0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
	} else {
		tmp = ((2.0 * (l * l)) / (k_m * (k_m * t))) / (k_m * k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d+50) then
        tmp = ((2.0d0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m))
    else
        tmp = ((2.0d0 * (l * l)) / (k_m * (k_m * t))) / (k_m * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+50) {
		tmp = ((2.0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
	} else {
		tmp = ((2.0 * (l * l)) / (k_m * (k_m * t))) / (k_m * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.1e+50:
		tmp = ((2.0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m))
	else:
		tmp = ((2.0 * (l * l)) / (k_m * (k_m * t))) / (k_m * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e+50)
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(Float64(k_m * k_m) * t)) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) / Float64(k_m * Float64(k_m * t))) / Float64(k_m * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e+50)
		tmp = ((2.0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
	else
		tmp = ((2.0 * (l * l)) / (k_m * (k_m * t))) / (k_m * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.1e+50], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000008e50

   1. Initial program 35.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 66.2

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{t \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

      10. *-lowering-*.f64 78.0

          \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   7. Applied egg-rr 78.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

   if 1.10000000000000008e50 < k

   1. Initial program 33.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

      9. pow-sqr N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      14. *-lowering-*.f64 55.2

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

      7. *-commutative N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      11. *-lowering-*.f64 56.4

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}}{k \cdot k}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}}{k \cdot k}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot k\right)}}}{k \cdot k} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 \cdot \left(\ell \cdot \ell\right)}}{t \cdot \left(k \cdot k\right)}}{k \cdot k} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot \left(k \cdot k\right)}}{k \cdot k} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(k \cdot k\right) \cdot t}}}{k \cdot k} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{k \cdot k} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{k \cdot k} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \color{blue}{\left(k \cdot t\right)}}}{k \cdot k} \]

      13. *-lowering-*.f64 61.0

          \[\leadsto \frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)}}{\color{blue}{k \cdot k}} \]

   9. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)}}{k \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2 \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* 2.0 l) (* (* k_m k_m) t)) (/ l (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((2.0 * l) / ((k_m * k_m) * t)) * (l / (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 = ((2.0d0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m))
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) * t)) * (l / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((2.0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(2.0 * l) / Float64(Float64(k_m * k_m) * t)) * Float64(l / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 * l) / ((k_m * k_m) * t)) * (l / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. *-lowering-*.f64 63.9

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 63.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

   10. *-lowering-*.f64 73.4

       \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

7. Applied egg-rr 73.4%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

8. Final simplification 73.4%

   \[\leadsto \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
9. Add Preprocessing

Alternative 18: 72.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* 2.0 l) (/ (/ l (* k_m (* k_m t))) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 * l) * ((l / (k_m * (k_m * t))) / (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 = (2.0d0 * l) * ((l / (k_m * (k_m * t))) / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * l) * ((l / (k_m * (k_m * t))) / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * l) * ((l / (k_m * (k_m * t))) / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * l) * Float64(Float64(l / Float64(k_m * Float64(k_m * t))) / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * l) * ((l / (k_m * (k_m * t))) / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. *-lowering-*.f64 63.9

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 63.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

   7. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

   11. *-lowering-*.f64 71.6

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

7. Applied egg-rr 71.6%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. associate-/r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{\ell}{t \cdot \left(k \cdot k\right)}}}{k \cdot k} \]

   5. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}}}{k \cdot k} \]

   6. associate-*l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{k \cdot k} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{k \cdot k} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}}}{k \cdot k} \]

   9. *-lowering-*.f64 72.8

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{\color{blue}{k \cdot k}} \]

9. Applied egg-rr 72.8%

   \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k}} \]
10. Add Preprocessing

Alternative 19: 70.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(2 \cdot \ell\right) \cdot \frac{\ell}{k\_m \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* 2.0 l) (/ 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) * (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) * (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) * (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) * (l / (k_m * ((k_m * k_m) * (k_m * t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * l) * Float64(l / Float64(k_m * Float64(Float64(k_m * k_m) * Float64(k_m * t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * l) * (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[(N[(2.0 * l), $MachinePrecision] * N[(l / N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. *-lowering-*.f64 63.9

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 63.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

   7. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

   11. *-lowering-*.f64 71.6

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

7. Applied egg-rr 71.6%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot k\right)} \]

   3. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot t\right)\right) \cdot k}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot t\right)\right) \cdot k}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot t\right)\right)} \cdot k} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot t\right)\right) \cdot k} \]

   7. *-lowering-*.f64 71.6

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot k} \]

9. Applied egg-rr 71.6%

   \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot t\right)\right) \cdot k}} \]

10. Final simplification 71.6%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot t\right)\right)} \]
11. Add Preprocessing

Alternative 20: 70.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* 2.0 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) {
	return (2.0 * l) * (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) * (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) * (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) * (l / ((k_m * k_m) * ((k_m * k_m) * t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * l) * Float64(l / Float64(Float64(k_m * k_m) * Float64(Float64(k_m * k_m) * t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 * l) * (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[(N[(2.0 * l), $MachinePrecision] * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. *-lowering-*.f64 63.9

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 63.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-*r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

   7. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

   11. *-lowering-*.f64 71.6

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

7. Applied egg-rr 71.6%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

8. Final simplification 71.6%

   \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))