Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 94.9%

Time: 16.2s

Alternatives: 20

Speedup: 11.0×

• 40.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)} \cdot \frac{2}{t}\right) \cdot \frac{\ell \cdot \cos k\_m}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    (* (/ l (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5))) (/ 2.0 t))
    (/ (* l (cos k_m)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((l / (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))) * (2.0 / t)) * ((l * cos(k_m)) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(l / Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5))) * Float64(2.0 / t)) * Float64(Float64(l * cos(k_m)) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. *-commutative N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 99.4%

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

Alternative 2: 91.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \left(\frac{2}{k\_m \cdot \mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right)} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    (/ (* l (cos k_m)) k_m)
    (* (/ 2.0 (* k_m (fma (cos (+ k_m k_m)) -0.5 0.5))) (/ l t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((l * cos(k_m)) / k_m) * ((2.0 / (k_m * fma(cos((k_m + k_m)), -0.5, 0.5))) * (l / t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(2.0 / Float64(k_m * fma(cos(Float64(k_m + k_m)), -0.5, 0.5))) * Float64(l / t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 / N[(k$95$m * N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 96.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k \cdot k\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k \cdot k}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \left(\frac{2}{k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \cdot \frac{\ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{2 \cdot \ell}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    (/ (* l (cos k_m)) k_m)
    (/ (* 2.0 l) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((l * cos(k_m)) / k_m) * ((2.0 * l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * (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 (k_m <= 1.15d-19) then
        tmp = (((t * (k_m * k_m)) / (2.0d0 * l)) ** (-1.0d0)) * (((k_m * k_m) / l) ** (-1.0d0))
    else
        tmp = ((l * cos(k_m)) / k_m) * ((2.0d0 * l) / ((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * (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 (k_m <= 1.15e-19) {
		tmp = Math.pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * Math.pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((l * Math.cos(k_m)) / k_m) * ((2.0 * l) / ((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.15e-19:
		tmp = math.pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * math.pow(((k_m * k_m) / l), -1.0)
	else:
		tmp = ((l * math.cos(k_m)) / k_m) * ((2.0 * l) / ((0.5 - (math.cos((k_m + k_m)) * 0.5)) * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(2.0 * l) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.15e-19)
		tmp = (((t * (k_m * k_m)) / (2.0 * l)) ^ -1.0) * (((k_m * k_m) / l) ^ -1.0);
	else
		tmp = ((l * cos(k_m)) / k_m) * ((2.0 * l) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k \cdot k\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k \cdot k}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{2 \cdot \ell}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k\_m} \cdot \frac{\ell \cdot \cos k\_m}{\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    (/ (* 2.0 l) k_m)
    (/ (* l (cos k_m)) (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (* k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((2.0 * l) / k_m) * ((l * cos(k_m)) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * (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 (k_m <= 1.15d-19) then
        tmp = (((t * (k_m * k_m)) / (2.0d0 * l)) ** (-1.0d0)) * (((k_m * k_m) / l) ** (-1.0d0))
    else
        tmp = ((2.0d0 * l) / k_m) * ((l * cos(k_m)) / ((0.5d0 - (cos((k_m + k_m)) * 0.5d0)) * (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 (k_m <= 1.15e-19) {
		tmp = Math.pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * Math.pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((2.0 * l) / k_m) * ((l * Math.cos(k_m)) / ((0.5 - (Math.cos((k_m + k_m)) * 0.5)) * (k_m * t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.15e-19:
		tmp = math.pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * math.pow(((k_m * k_m) / l), -1.0)
	else:
		tmp = ((2.0 * l) / k_m) * ((l * math.cos(k_m)) / ((0.5 - (math.cos((k_m + k_m)) * 0.5)) * (k_m * t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k_m) * Float64(Float64(l * cos(k_m)) / Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m * t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.15e-19)
		tmp = (((t * (k_m * k_m)) / (2.0 * l)) ^ -1.0) * (((k_m * k_m) / l) ^ -1.0);
	else
		tmp = ((2.0 * l) / k_m) * ((l * cos(k_m)) / ((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m * t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.0

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k \cdot k\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k \cdot k}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell \cdot \cos k}{\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \left(\ell \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    (/ (* l (cos k_m)) k_m)
    (* l (/ 2.0 (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = ((l * cos(k_m)) / k_m) * (l * (2.0 / (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(l * Float64(2.0 / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l * N[(2.0 / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

   9. Applied rewrites 93.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k \cdot k\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k \cdot k}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \left(\ell \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\cos k\_m}{k\_m} \cdot \frac{2 \cdot \ell}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    l
    (*
     (/ (cos k_m) k_m)
     (/ (* 2.0 l) (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = l * ((cos(k_m) / k_m) * ((2.0 * l) / (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(l * Float64(Float64(cos(k_m) / k_m) * Float64(Float64(2.0 * l) / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied rewrites 90.1%

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

Alternative 7: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;{\left(\frac{t \cdot \left(k\_m \cdot k\_m\right)}{2 \cdot \ell}\right)}^{-1} \cdot {\left(\frac{k\_m \cdot k\_m}{\ell}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell \cdot \cos k\_m}{k\_m \cdot \left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.15e-19)
   (* (pow (/ (* t (* k_m k_m)) (* 2.0 l)) -1.0) (pow (/ (* k_m k_m) l) -1.0))
   (*
    (* 2.0 l)
    (/
     (* l (cos k_m))
     (* k_m (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.15e-19) {
		tmp = pow(((t * (k_m * k_m)) / (2.0 * l)), -1.0) * pow(((k_m * k_m) / l), -1.0);
	} else {
		tmp = (2.0 * l) * ((l * cos(k_m)) / (k_m * (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.15e-19)
		tmp = Float64((Float64(Float64(t * Float64(k_m * k_m)) / Float64(2.0 * l)) ^ -1.0) * (Float64(Float64(k_m * k_m) / l) ^ -1.0));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(l * cos(k_m)) / Float64(k_m * Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.15e-19], N[(N[Power[N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(2.0 * l), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. unpow-prod-down N/A

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

      18. lower-*.f64 N/A

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

   7. Applied rewrites 81.2%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 80.6

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

   5. Applied rewrites 80.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

   9. Applied rewrites 84.7%

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

Alternative 8: 76.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{elif}\;k\_m \leq 2.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\ell \cdot \cos k\_m}{k\_m} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k\_m \cdot k\_m, 2\right)}{k\_m \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)} \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= k_m 2.1e-73)
     (* (/ 2.0 t) (* t_1 t_1))
     (if (<= k_m 2.8e+78)
       (*
        (/ (* l (cos k_m)) k_m)
        (/
         (* (/ l t) (fma 0.6666666666666666 (* k_m k_m) 2.0))
         (* k_m (* k_m k_m))))
       (*
        (/ (* 2.0 l) (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t)))
        (/ l k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (k_m <= 2.1e-73) {
		tmp = (2.0 / t) * (t_1 * t_1);
	} else if (k_m <= 2.8e+78) {
		tmp = ((l * cos(k_m)) / k_m) * (((l / t) * fma(0.6666666666666666, (k_m * k_m), 2.0)) / (k_m * (k_m * k_m)));
	} else {
		tmp = ((2.0 * l) / (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t))) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 2.1e-73)
		tmp = Float64(Float64(2.0 / t) * Float64(t_1 * t_1));
	elseif (k_m <= 2.8e+78)
		tmp = Float64(Float64(Float64(l * cos(k_m)) / k_m) * Float64(Float64(Float64(l / t) * fma(0.6666666666666666, Float64(k_m * k_m), 2.0)) / Float64(k_m * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t))) * Float64(l / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 2.1e-73], N[(N[(2.0 / t), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.8e+78], N[(N[(N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * N[(0.6666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.0999999999999999e-73

   1. Initial program 39.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. cube-unmult N/A

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

      14. lower-*.f64 N/A

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

      15. cube-unmult N/A

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

      16. lift-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 66.7

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

   7. Applied rewrites 66.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 79.1

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

   9. Applied rewrites 79.1%

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

   if 2.0999999999999999e-73 < k < 2.8000000000000001e78

   1. Initial program 40.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 88.2

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

   5. Applied rewrites 88.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 80.5%

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

   if 2.8000000000000001e78 < k

   1. Initial program 36.1%

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

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 63.4

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

   8. Applied rewrites 63.4%

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

   10. Applied rewrites 66.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{t} \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\ell \cdot \cos k}{k} \cdot \frac{\frac{\ell}{t} \cdot \mathsf{fma}\left(0.6666666666666666, k \cdot k, 2\right)}{k \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 30:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)} \cdot \frac{\ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 30.0)
   (* (/ l (* k_m k_m)) (/ (* 2.0 l) (* t (* k_m k_m))))
   (* (/ (* 2.0 l) (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t))) (/ l k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 30.0) {
		tmp = (l / (k_m * k_m)) * ((2.0 * l) / (t * (k_m * k_m)));
	} else {
		tmp = ((2.0 * l) / (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t))) * (l / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 30.0)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t))) * Float64(l / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 30.0], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 30

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if 30 < k

   1. Initial program 36.1%

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

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

   10. Applied rewrites 65.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 30:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 30:\\ \;\;\;\;\frac{\ell}{k\_m \cdot k\_m} \cdot \frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell}{k\_m \cdot \left(\mathsf{fma}\left(\cos \left(k\_m + k\_m\right), -0.5, 0.5\right) \cdot \left(k\_m \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 30.0)
   (* (/ l (* k_m k_m)) (/ (* 2.0 l) (* t (* k_m k_m))))
   (* (* 2.0 l) (/ l (* k_m (* (fma (cos (+ k_m k_m)) -0.5 0.5) (* k_m t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 30.0) {
		tmp = (l / (k_m * k_m)) * ((2.0 * l) / (t * (k_m * k_m)));
	} else {
		tmp = (2.0 * l) * (l / (k_m * (fma(cos((k_m + k_m)), -0.5, 0.5) * (k_m * t))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 30.0)
		tmp = Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(l / Float64(k_m * Float64(fma(cos(Float64(k_m + k_m)), -0.5, 0.5) * Float64(k_m * t)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 30.0], N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(l / N[(k$95$m * N[(N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 30

   1. Initial program 40.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if 30 < k

   1. Initial program 36.1%

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

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-sin.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 62.8

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

   8. Applied rewrites 62.8%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 64.8

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. sqr-sin-a N/A

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

      20. count-2 N/A

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

      21. lift-+.f64 N/A

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

   10. Applied rewrites 64.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 30:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-148}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell}{k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell \cdot \ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1e-148)
   (* (* 2.0 l) (/ l (* k_m (* t (* k_m (* k_m k_m))))))
   (* (/ 2.0 (* t (* k_m k_m))) (/ (* l l) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1e-148) {
		tmp = (2.0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))));
	} else {
		tmp = (2.0 / (t * (k_m * k_m))) * ((l * l) / (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 (l <= 1d-148) then
        tmp = (2.0d0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))))
    else
        tmp = (2.0d0 / (t * (k_m * k_m))) * ((l * l) / (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 (l <= 1e-148) {
		tmp = (2.0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))));
	} else {
		tmp = (2.0 / (t * (k_m * k_m))) * ((l * l) / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1e-148:
		tmp = (2.0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))))
	else:
		tmp = (2.0 / (t * (k_m * k_m))) * ((l * l) / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 1e-148)
		tmp = Float64(Float64(2.0 * l) * Float64(l / Float64(k_m * Float64(t * Float64(k_m * Float64(k_m * k_m))))));
	else
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k_m * k_m))) * Float64(Float64(l * l) / 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 (l <= 1e-148)
		tmp = (2.0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))));
	else
		tmp = (2.0 / (t * (k_m * k_m))) * ((l * l) / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1e-148], N[(N[(2.0 * l), $MachinePrecision] * N[(l / N[(k$95$m * N[(t * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.99999999999999936e-149

   1. Initial program 38.9%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 73.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. cube-unmult N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. cube-unmult N/A

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

      21. lift-*.f64 N/A

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

      22. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 9.99999999999999936e-149 < l

   1. Initial program 39.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 70.3

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

   7. Applied rewrites 70.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-148}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\frac{2}{t} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* (/ 2.0 t) (/ l k_m)) (/ l (* k_m (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((2.0 / t) * (l / k_m)) * (l / (k_m * (k_m * k_m)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((2.0d0 / t) * (l / k_m)) * (l / (k_m * (k_m * k_m)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((2.0 / t) * (l / k_m)) * (l / (k_m * (k_m * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((2.0 / t) * (l / k_m)) * (l / (k_m * (k_m * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(2.0 / t) * Float64(l / k_m)) * Float64(l / Float64(k_m * Float64(k_m * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((2.0 / t) * (l / k_m)) * (l / (k_m * (k_m * k_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(2.0 / t), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. cube-unmult N/A

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

   14. lower-*.f64 N/A

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

   15. cube-unmult N/A

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

   16. lift-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 65.1

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

7. Applied rewrites 65.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. times-frac N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 74.9

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

9. Applied rewrites 74.9%

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

Alternative 13: 72.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{k\_m \cdot k\_m} \cdot \frac{2 \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (* k_m k_m)) (/ (* 2.0 l) (* t (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (k_m * k_m)) * ((2.0 * l) / (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 = (l / (k_m * k_m)) * ((2.0d0 * l) / (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 (l / (k_m * k_m)) * ((2.0 * l) / (t * (k_m * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / (k_m * k_m)) * ((2.0 * l) / (t * (k_m * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(k_m * k_m)) * Float64(Float64(2.0 * l) / Float64(t * Float64(k_m * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / (k_m * k_m)) * ((2.0 * l) / (t * (k_m * k_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 74.9

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

7. Applied rewrites 74.9%

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

8. Final simplification 74.9%

   \[\leadsto \frac{\ell}{k \cdot k} \cdot \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \]
9. Add Preprocessing

Alternative 14: 69.0% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(2 \cdot \ell\right) \cdot \frac{\ell}{k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* 2.0 l) (/ l (* k_m (* t (* k_m (* 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 * (t * (k_m * (k_m * k_m)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 * l) * (l / (k_m * (t * (k_m * (k_m * k_m)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 * l) * Float64(l / Float64(k_m * Float64(t * Float64(k_m * 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 * (t * (k_m * (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[(l / N[(k$95$m * N[(t * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

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

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 70.3

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. cube-unmult N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. cube-unmult N/A

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

   21. lift-*.f64 N/A

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

   22. lower-*.f64 72.3

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

7. Applied rewrites 72.3%

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

8. Final simplification 72.3%

   \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)} \]
9. Add Preprocessing

Alternative 15: 20.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ k\_m \cdot \frac{k\_m}{\frac{t}{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* k_m (/ k_m (/ t (* (* l l) -0.0205026455026455)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return k_m * (k_m / (t / ((l * l) * -0.0205026455026455)));
}

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 = k_m * (k_m / (t / ((l * l) * (-0.0205026455026455d0))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return k_m * (k_m / (t / ((l * l) * -0.0205026455026455)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return k_m * (k_m / (t / ((l * l) * -0.0205026455026455)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(k_m * Float64(k_m / Float64(t / Float64(Float64(l * l) * -0.0205026455026455))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = k_m * (k_m / (t / ((l * l) * -0.0205026455026455)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(k$95$m * N[(k$95$m / N[(t / N[(N[(l * l), $MachinePrecision] * -0.0205026455026455), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-31}{1512}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-31}{1512} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-31}{1512}\right)} \]

   4. *-commutative N/A

      \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   11. *-commutative N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   13. unpow2 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   14. lower-*.f64 20.2

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.0205026455026455}{t}\right) \]

8. Applied rewrites 20.2%

   \[\leadsto \color{blue}{k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{-31}{1512}}}{t}\right) \]

   3. clear-num N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{1}{\frac{t}{\left(\ell \cdot \ell\right) \cdot \frac{-31}{1512}}}}\right) \]

   4. un-div-inv N/A

      \[\leadsto k \cdot \color{blue}{\frac{k}{\frac{t}{\left(\ell \cdot \ell\right) \cdot \frac{-31}{1512}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto k \cdot \color{blue}{\frac{k}{\frac{t}{\left(\ell \cdot \ell\right) \cdot \frac{-31}{1512}}}} \]

   6. lower-/.f64 20.7

      \[\leadsto k \cdot \frac{k}{\color{blue}{\frac{t}{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}}} \]

10. Applied rewrites 20.7%

    \[\leadsto k \cdot \color{blue}{\frac{k}{\frac{t}{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}}} \]
11. Add Preprocessing

Alternative 16: 19.6% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ k\_m \cdot \left(k\_m \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* k_m (* k_m (/ (* (* l l) -0.0205026455026455) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return k_m * (k_m * (((l * l) * -0.0205026455026455) / 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 = k_m * (k_m * (((l * l) * (-0.0205026455026455d0)) / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return k_m * (k_m * (((l * l) * -0.0205026455026455) / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return k_m * (k_m * (((l * l) * -0.0205026455026455) / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(k_m * Float64(k_m * Float64(Float64(Float64(l * l) * -0.0205026455026455) / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = k_m * (k_m * (((l * l) * -0.0205026455026455) / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(k$95$m * N[(k$95$m * N[(N[(N[(l * l), $MachinePrecision] * -0.0205026455026455), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-31}{1512}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-31}{1512} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-31}{1512}\right)} \]

   4. *-commutative N/A

      \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   11. *-commutative N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   13. unpow2 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   14. lower-*.f64 20.2

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.0205026455026455}{t}\right) \]

8. Applied rewrites 20.2%

   \[\leadsto \color{blue}{k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right)} \]
9. Add Preprocessing

Alternative 17: 17.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ k\_m \cdot \left(-0.0205026455026455 \cdot \frac{k\_m \cdot \left(\ell \cdot \ell\right)}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* k_m (* -0.0205026455026455 (/ (* k_m (* l l)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return k_m * (-0.0205026455026455 * ((k_m * (l * l)) / t));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = k_m * ((-0.0205026455026455d0) * ((k_m * (l * l)) / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return k_m * (-0.0205026455026455 * ((k_m * (l * l)) / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return k_m * (-0.0205026455026455 * ((k_m * (l * l)) / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(k_m * Float64(-0.0205026455026455 * Float64(Float64(k_m * Float64(l * l)) / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = k_m * (-0.0205026455026455 * ((k_m * (l * l)) / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(k$95$m * N[(-0.0205026455026455 * N[(N[(k$95$m * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-31}{1512}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-31}{1512} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-31}{1512}\right)} \]

   4. *-commutative N/A

      \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   11. *-commutative N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   13. unpow2 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   14. lower-*.f64 20.2

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.0205026455026455}{t}\right) \]

8. Applied rewrites 20.2%

   \[\leadsto \color{blue}{k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\frac{k \cdot {\ell}^{2}}{t}}\right) \]

    3. *-commutative N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    5. unpow2 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

    6. lower-*.f64 19.1

       \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

11. Applied rewrites 19.1%

    \[\leadsto k \cdot \color{blue}{\left(-0.0205026455026455 \cdot \frac{\left(\ell \cdot \ell\right) \cdot k}{t}\right)} \]

12. Final simplification 19.1%

    \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \frac{k \cdot \left(\ell \cdot \ell\right)}{t}\right) \]
13. Add Preprocessing

Alternative 18: 12.6% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ k\_m \cdot \left(-0.0205026455026455 \cdot \left(\ell \cdot \frac{k\_m \cdot \ell}{t}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* k_m (* -0.0205026455026455 (* l (/ (* k_m l) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return k_m * (-0.0205026455026455 * (l * ((k_m * l) / t)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = k_m * ((-0.0205026455026455d0) * (l * ((k_m * l) / t)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return k_m * (-0.0205026455026455 * (l * ((k_m * l) / t)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return k_m * (-0.0205026455026455 * (l * ((k_m * l) / t)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(k_m * Float64(-0.0205026455026455 * Float64(l * Float64(Float64(k_m * l) / t))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = k_m * (-0.0205026455026455 * (l * ((k_m * l) / t)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(k$95$m * N[(-0.0205026455026455 * N[(l * N[(N[(k$95$m * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-31}{1512}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-31}{1512} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-31}{1512}\right)} \]

   4. *-commutative N/A

      \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   11. *-commutative N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   13. unpow2 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   14. lower-*.f64 20.2

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.0205026455026455}{t}\right) \]

8. Applied rewrites 20.2%

   \[\leadsto \color{blue}{k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\frac{k \cdot {\ell}^{2}}{t}}\right) \]

    3. *-commutative N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    5. unpow2 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

    6. lower-*.f64 19.1

       \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

11. Applied rewrites 19.1%

    \[\leadsto k \cdot \color{blue}{\left(-0.0205026455026455 \cdot \frac{\left(\ell \cdot \ell\right) \cdot k}{t}\right)} \]
12. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot k\right)}}{t}\right) \]

    2. associate-/l* N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\left(\ell \cdot \frac{\ell \cdot k}{t}\right)}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\left(\ell \cdot \frac{\ell \cdot k}{t}\right)}\right) \]

    4. lower-/.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \left(\ell \cdot \color{blue}{\frac{\ell \cdot k}{t}}\right)\right) \]

    5. lower-*.f64 10.8

       \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \left(\ell \cdot \frac{\color{blue}{\ell \cdot k}}{t}\right)\right) \]

13. Applied rewrites 10.8%

    \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \color{blue}{\left(\ell \cdot \frac{\ell \cdot k}{t}\right)}\right) \]

14. Final simplification 10.8%

    \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \left(\ell \cdot \frac{k \cdot \ell}{t}\right)\right) \]
15. Add Preprocessing

Alternative 19: 12.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ k\_m \cdot \left(-0.0205026455026455 \cdot \left(\ell \cdot \left(\ell \cdot \frac{k\_m}{t}\right)\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* k_m (* -0.0205026455026455 (* l (* l (/ k_m t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return k_m * (-0.0205026455026455 * (l * (l * (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 = k_m * ((-0.0205026455026455d0) * (l * (l * (k_m / t))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return k_m * (-0.0205026455026455 * (l * (l * (k_m / t))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return k_m * (-0.0205026455026455 * (l * (l * (k_m / t))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(k_m * Float64(-0.0205026455026455 * Float64(l * Float64(l * Float64(k_m / t)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = k_m * (-0.0205026455026455 * (l * (l * (k_m / t))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(k$95$m * N[(-0.0205026455026455 * N[(l * N[(l * N[(k$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-31}{1512}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-31}{1512} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-31}{1512}\right)} \]

   4. *-commutative N/A

      \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   11. *-commutative N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   13. unpow2 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   14. lower-*.f64 20.2

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.0205026455026455}{t}\right) \]

8. Applied rewrites 20.2%

   \[\leadsto \color{blue}{k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\frac{k \cdot {\ell}^{2}}{t}}\right) \]

    3. *-commutative N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    5. unpow2 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

    6. lower-*.f64 19.1

       \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

11. Applied rewrites 19.1%

    \[\leadsto k \cdot \color{blue}{\left(-0.0205026455026455 \cdot \frac{\left(\ell \cdot \ell\right) \cdot k}{t}\right)} \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

    2. associate-/l* N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \frac{k}{t}\right)}\right) \]

    3. lift-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{k}{t}\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{k}{t}\right)\right)}\right) \]

    5. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{k}{t}\right)\right)}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{k}{t}\right)}\right)\right) \]

    7. lower-/.f64 10.8

       \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{k}{t}}\right)\right)\right) \]

13. Applied rewrites 10.8%

    \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \frac{k}{t}\right)\right)}\right) \]
14. Add Preprocessing

Alternative 20: 9.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ -0.0205026455026455 \cdot \left(\ell \cdot \frac{\left(k\_m \cdot k\_m\right) \cdot \ell}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* -0.0205026455026455 (* l (/ (* (* k_m k_m) l) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return -0.0205026455026455 * (l * (((k_m * k_m) * l) / t));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (-0.0205026455026455d0) * (l * (((k_m * k_m) * l) / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return -0.0205026455026455 * (l * (((k_m * k_m) * l) / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return -0.0205026455026455 * (l * (((k_m * k_m) * l) / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(-0.0205026455026455 * Float64(l * Float64(Float64(Float64(k_m * k_m) * l) / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = -0.0205026455026455 * (l * (((k_m * k_m) * l) / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(-0.0205026455026455 * N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 23.0%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-31}{1512}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right)} \cdot \frac{-31}{1512} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-31}{1512}\right)} \]

   4. *-commutative N/A

      \[\leadsto {k}^{2} \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)} \]

   5. unpow2 N/A

      \[\leadsto \color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*l* N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(\frac{-31}{1512} \cdot \frac{{\ell}^{2}}{t}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   10. lower-/.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \color{blue}{\frac{\frac{-31}{1512} \cdot {\ell}^{2}}{t}}\right) \]

   11. *-commutative N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{-31}{1512}}}{t}\right) \]

   13. unpow2 N/A

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-31}{1512}}{t}\right) \]

   14. lower-*.f64 20.2

       \[\leadsto k \cdot \left(k \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.0205026455026455}{t}\right) \]

8. Applied rewrites 20.2%

   \[\leadsto \color{blue}{k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.0205026455026455}{t}\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto k \cdot \color{blue}{\left(\frac{-31}{1512} \cdot \frac{k \cdot {\ell}^{2}}{t}\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \color{blue}{\frac{k \cdot {\ell}^{2}}{t}}\right) \]

    3. *-commutative N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot k}}{t}\right) \]

    5. unpow2 N/A

       \[\leadsto k \cdot \left(\frac{-31}{1512} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

    6. lower-*.f64 19.1

       \[\leadsto k \cdot \left(-0.0205026455026455 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot k}{t}\right) \]

11. Applied rewrites 19.1%

    \[\leadsto k \cdot \color{blue}{\left(-0.0205026455026455 \cdot \frac{\left(\ell \cdot \ell\right) \cdot k}{t}\right)} \]

12. Taylor expanded in k around 0

    \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-31}{1512} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-31}{1512} \cdot \frac{\color{blue}{{\ell}^{2} \cdot {k}^{2}}}{t} \]

    3. unpow2 N/A

       \[\leadsto \frac{-31}{1512} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot {k}^{2}}{t} \]

    4. associate-*l* N/A

       \[\leadsto \frac{-31}{1512} \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot {k}^{2}\right)}}{t} \]

    5. *-commutative N/A

       \[\leadsto \frac{-31}{1512} \cdot \frac{\ell \cdot \color{blue}{\left({k}^{2} \cdot \ell\right)}}{t} \]

    6. associate-/l* N/A

       \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{2} \cdot \ell}{t}\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{-31}{1512} \cdot \color{blue}{\left(\ell \cdot \frac{{k}^{2} \cdot \ell}{t}\right)} \]

    8. lower-/.f64 N/A

       \[\leadsto \frac{-31}{1512} \cdot \left(\ell \cdot \color{blue}{\frac{{k}^{2} \cdot \ell}{t}}\right) \]

    9. *-commutative N/A

       \[\leadsto \frac{-31}{1512} \cdot \left(\ell \cdot \frac{\color{blue}{\ell \cdot {k}^{2}}}{t}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{-31}{1512} \cdot \left(\ell \cdot \frac{\color{blue}{\ell \cdot {k}^{2}}}{t}\right) \]

    11. unpow2 N/A

        \[\leadsto \frac{-31}{1512} \cdot \left(\ell \cdot \frac{\ell \cdot \color{blue}{\left(k \cdot k\right)}}{t}\right) \]

    12. lower-*.f64 7.0

        \[\leadsto -0.0205026455026455 \cdot \left(\ell \cdot \frac{\ell \cdot \color{blue}{\left(k \cdot k\right)}}{t}\right) \]

14. Applied rewrites 7.0%

    \[\leadsto \color{blue}{-0.0205026455026455 \cdot \left(\ell \cdot \frac{\ell \cdot \left(k \cdot k\right)}{t}\right)} \]

15. Final simplification 7.0%

    \[\leadsto -0.0205026455026455 \cdot \left(\ell \cdot \frac{\left(k \cdot k\right) \cdot \ell}{t}\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))