Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 97.1%

Time: 13.1s

Alternatives: 13

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: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+275}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\sin k \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)}{\ell} \cdot \frac{t}{\cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 4e+275)
   (/ 2.0 (* (/ k l) (* (sin k) (* (sin k) (/ (* k t) (* l (cos k)))))))
   (/
    2.0
    (* (/ k l) (* (/ (* k (fma (cos (+ k k)) -0.5 0.5)) l) (/ t (cos k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 4e+275) {
		tmp = 2.0 / ((k / l) * (sin(k) * (sin(k) * ((k * t) / (l * cos(k))))));
	} else {
		tmp = 2.0 / ((k / l) * (((k * fma(cos((k + k)), -0.5, 0.5)) / l) * (t / cos(k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 4e+275)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(sin(k) * Float64(sin(k) * Float64(Float64(k * t) / Float64(l * cos(k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(Float64(k * fma(cos(Float64(k + k)), -0.5, 0.5)) / l) * Float64(t / cos(k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e+275], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[(k * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 3.99999999999999984e275

   1. Initial program 39.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 85.0

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

   5. Simplified 85.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 82.0%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. sqr-sin-a N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 94.8

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

   9. Applied egg-rr 94.8%

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

       1. lift-sin.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 98.8

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

   11. Applied egg-rr 98.8%

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

   if 3.99999999999999984e275 < (*.f64 l l)

   1. Initial program 36.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 67.9

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

   5. Simplified 67.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 89.9%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied egg-rr 99.3%

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

Alternative 2: 83.1% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-19)
   (/ 2.0 (* (* (/ k l) (* k k)) (/ (* k t) l)))
   (/
    2.0
    (* (/ k l) (* (/ (* k (fma (cos (+ k k)) -0.5 0.5)) l) (/ t (cos k)))))))

C

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

Julia

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

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-19], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[(k * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Cos[k], $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

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.9

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

   7. Applied egg-rr 81.9%

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

   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

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 79.8

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

   5. Simplified 79.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 93.1%

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

      1. lift-+.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-cos.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   9. Applied egg-rr 98.4%

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

4. Final simplification 86.4%

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

Alternative 3: 88.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-179)
   (/ (/ (/ 2.0 (* k k)) (/ k l)) (/ (* k t) l))
   (if (<= (* l l) 2e+305)
     (/ 2.0 (* k (* (* t (/ k (* l l))) (* (sin k) (tan k)))))
     (*
      l
      (/
       (* l (* 2.0 (cos k)))
       (* (fma (cos (+ k k)) -0.5 0.5) (* k (* k t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-179) {
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	} else if ((l * l) <= 2e+305) {
		tmp = 2.0 / (k * ((t * (k / (l * l))) * (sin(k) * tan(k))));
	} else {
		tmp = l * ((l * (2.0 * cos(k))) / (fma(cos((k + k)), -0.5, 0.5) * (k * (k * t))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-179)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(k / l)) / Float64(Float64(k * t) / l));
	elseif (Float64(l * l) <= 2e+305)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t * Float64(k / Float64(l * l))) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 * cos(k))) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * Float64(k * t)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-179], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+305], N[(2.0 / N[(k * N[(N[(t * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1e-179

   1. Initial program 31.9%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 71.0

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

   5. Simplified 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   if 1e-179 < (*.f64 l l) < 1.9999999999999999e305

   1. Initial program 46.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 94.3

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

   5. Simplified 94.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 99.7%

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

   if 1.9999999999999999e305 < (*.f64 l l)

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 66.9

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

   5. Simplified 66.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 88.3%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos k\right) \cdot \left(\ell \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-179)
   (/ (/ (/ 2.0 (* k k)) (/ k l)) (/ (* k t) l))
   (if (<= (* l l) 2e+305)
     (/ 2.0 (* k (* (* t (/ k (* l l))) (* (sin k) (tan k)))))
     (*
      (* l (cos k))
      (* l (/ 2.0 (* (fma (cos (+ k k)) -0.5 0.5) (* k (* k t)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-179) {
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	} else if ((l * l) <= 2e+305) {
		tmp = 2.0 / (k * ((t * (k / (l * l))) * (sin(k) * tan(k))));
	} else {
		tmp = (l * cos(k)) * (l * (2.0 / (fma(cos((k + k)), -0.5, 0.5) * (k * (k * t)))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-179)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(k / l)) / Float64(Float64(k * t) / l));
	elseif (Float64(l * l) <= 2e+305)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t * Float64(k / Float64(l * l))) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(Float64(l * cos(k)) * Float64(l * Float64(2.0 / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * Float64(k * t))))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-179], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+305], N[(2.0 / N[(k * N[(N[(t * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1e-179

   1. Initial program 31.9%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 71.0

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

   5. Simplified 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   if 1e-179 < (*.f64 l l) < 1.9999999999999999e305

   1. Initial program 46.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 94.3

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

   5. Simplified 94.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 99.7%

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

   if 1.9999999999999999e305 < (*.f64 l l)

   1. Initial program 35.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 66.9

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

   5. Simplified 66.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 88.3%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos k\right) \cdot \left(\ell \cdot \frac{2}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-19)
   (/ 2.0 (* (* (/ k l) (* k k)) (/ (* k t) l)))
   (*
    (* 2.0 (/ l k))
    (* (/ (cos k) k) (/ l (* t (fma (cos (+ k k)) -0.5 0.5)))))))

C

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

Julia

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

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-19], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $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

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.9

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

   7. Applied egg-rr 81.9%

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

   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

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 79.8

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

   5. Simplified 79.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 93.1%

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

   9. Applied egg-rr 93.9%

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

       1. lift-cos.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-+.f64 N/A

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

       4. lift-cos.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 97.0

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

   11. Applied egg-rr 97.0%

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

4. Final simplification 86.1%

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

Alternative 6: 81.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.15e-19)
   (/ 2.0 (* (* (/ k l) (* k k)) (/ (* k t) l)))
   (*
    (* 2.0 (/ l k))
    (* l (/ (cos k) (* (* k t) (fma (cos (+ k k)) -0.5 0.5)))))))

C

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

Julia

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

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.15e-19], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $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

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 72.1

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

   5. Simplified 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 81.9

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

   7. Applied egg-rr 81.9%

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

   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

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 79.8

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

   5. Simplified 79.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied egg-rr 93.1%

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

   9. Applied egg-rr 93.9%

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

       1. lift-cos.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

   11. Applied egg-rr 94.0%

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

4. Final simplification 85.2%

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

Alternative 7: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-179)
   (/ (/ (/ 2.0 (* k k)) (/ k l)) (/ (* k t) l))
   (/ 2.0 (* k (* (* t (/ k (* l l))) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-179) {
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	} else {
		tmp = 2.0 / (k * ((t * (k / (l * l))) * (sin(k) * tan(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-179) then
        tmp = ((2.0d0 / (k * k)) / (k / l)) / ((k * t) / l)
    else
        tmp = 2.0d0 / (k * ((t * (k / (l * l))) * (sin(k) * tan(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-179) {
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	} else {
		tmp = 2.0 / (k * ((t * (k / (l * l))) * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-179:
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l)
	else:
		tmp = 2.0 / (k * ((t * (k / (l * l))) * (math.sin(k) * math.tan(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-179)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(k / l)) / Float64(Float64(k * t) / l));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t * Float64(k / Float64(l * l))) * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-179)
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	else
		tmp = 2.0 / (k * ((t * (k / (l * l))) * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-179], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(t * N[(k / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-179

   1. Initial program 31.9%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 71.0

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

   5. Simplified 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   if 1e-179 < (*.f64 l l)

   1. Initial program 42.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 83.5

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

   5. Simplified 83.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied egg-rr 87.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-179)
   (/ (/ (/ 2.0 (* k k)) (/ k l)) (/ (* k t) l))
   (/ 2.0 (* k (/ (* k (* k (* k t))) (* (* l l) (cos k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-179) {
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	} else {
		tmp = 2.0 / (k * ((k * (k * (k * t))) / ((l * l) * cos(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-179) then
        tmp = ((2.0d0 / (k * k)) / (k / l)) / ((k * t) / l)
    else
        tmp = 2.0d0 / (k * ((k * (k * (k * t))) / ((l * l) * cos(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-179) {
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	} else {
		tmp = 2.0 / (k * ((k * (k * (k * t))) / ((l * l) * Math.cos(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-179:
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l)
	else:
		tmp = 2.0 / (k * ((k * (k * (k * t))) / ((l * l) * math.cos(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-179)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * k)) / Float64(k / l)) / Float64(Float64(k * t) / l));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(k * Float64(k * Float64(k * t))) / Float64(Float64(l * l) * cos(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-179)
		tmp = ((2.0 / (k * k)) / (k / l)) / ((k * t) / l);
	else
		tmp = 2.0 / (k * ((k * (k * (k * t))) / ((l * l) * cos(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-179], N[(N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-179

   1. Initial program 31.9%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 71.0

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

   5. Simplified 71.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 94.6

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

   7. Applied egg-rr 94.6%

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

   if 1e-179 < (*.f64 l l)

   1. Initial program 42.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-cos.f64 83.5

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

   5. Simplified 83.5%

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

   6. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.2

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

   8. Simplified 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 69.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 70.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 70.0

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

   10. Applied egg-rr 70.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-179}:\\ \;\;\;\;\frac{\frac{\frac{2}{k \cdot k}}{\frac{k}{\ell}}}{\frac{k \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ k l) (* k k)) (/ (* k t) l))))

C

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

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 / (((k / l) * (k * k)) * ((k * t) / l))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k / l) * (k * k)) * ((k * t) / l));
}

Python

def code(t, l, k):
	return 2.0 / (((k / l) * (k * k)) * ((k * t) / l))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $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

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

   1. associate-/l* N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 68.9

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

5. Simplified 68.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. times-frac N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 75.7

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

7. Applied egg-rr 75.7%

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

8. Final simplification 75.7%

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

Alternative 10: 72.1% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* k k) (* (/ k l) (/ (* k t) l)))))

C

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

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 / ((k * k) * ((k / l) * ((k * t) / l)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((k * k) * ((k / l) * ((k * t) / l)));
}

Python

def code(t, l, k):
	return 2.0 / ((k * k) * ((k / l) * ((k * t) / l)))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / l), $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

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

   1. associate-/l* N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 68.9

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

5. Simplified 68.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 75.3

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

7. Applied egg-rr 75.3%

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

8. Final simplification 75.3%

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

Alternative 11: 70.4% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l (* k t)) (* 2.0 (/ l (* k (* k k))))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return (l / (k * t)) * (2.0 * (l / (k * (k * k))));
}

Python

def code(t, l, k):
	return (l / (k * t)) * (2.0 * (l / (k * (k * k))))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(l / N[(k * N[(k * k), $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

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

   1. associate-/l* N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 68.9

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

5. Simplified 68.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 68.0

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

7. Applied egg-rr 68.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 68.0

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. cube-mult N/A

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

   16. lower-*.f64 N/A

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

   17. cube-mult N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 67.6

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

9. Applied egg-rr 67.6%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-/.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. times-frac N/A

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

    12. associate-*l* N/A

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

    13. lower-*.f64 N/A

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

    14. lower-/.f64 N/A

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

    15. lower-*.f64 N/A

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

    16. lower-/.f64 74.9

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

11. Applied egg-rr 74.9%

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

12. Final simplification 74.9%

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

Alternative 12: 72.2% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ l (* t (* k k))) (/ l (* k k)))))

C

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

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 * ((l / (t * (k * k))) * (l / (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / (t * (k * k))) * (l / (k * k)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / (t * (k * k))) * (l / (k * k)))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $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

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

   1. associate-/l* N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 68.9

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

5. Simplified 68.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 68.0

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

7. Applied egg-rr 68.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   9. lower-/.f64 68.0

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   12. *-commutative N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \left(k \cdot k\right)} \]

   13. associate-*l* N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   15. cube-mult N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{{k}^{3}}} \]

   16. lower-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot {k}^{3}}} \]

   17. cube-mult N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

   18. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   19. lower-*.f64 67.6

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

9. Applied egg-rr 67.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(k \cdot k\right)\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

    3. associate-*r* N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(k \cdot k\right)}} \]

    4. *-commutative N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

    5. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

    6. times-frac N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k \cdot k}\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k \cdot k}\right)} \]

    8. lower-/.f64 N/A

       \[\leadsto 2 \cdot \left(\color{blue}{\frac{\ell}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot k}\right) \]

    9. lift-*.f64 N/A

       \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot k}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot k}\right) \]

    11. associate-*r* N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \cdot \frac{\ell}{k \cdot k}\right) \]

    12. lift-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell}{k \cdot k}\right) \]

    13. *-commutative N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k}\right) \]

    14. lower-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k}\right) \]

    15. lower-/.f64 74.9

        \[\leadsto 2 \cdot \left(\frac{\ell}{t \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

11. Applied egg-rr 74.9%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\right)} \]
12. Add Preprocessing

Alternative 13: 69.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* l (/ l (* k (* k (* t (* k k))))))))

C

double code(double t, double l, double k) {
	return 2.0 * (l * (l / (k * (k * (t * (k * k))))));
}

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 * (l * (l / (k * (k * (t * (k * k))))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (l * (l / (k * (k * (t * (k * k))))));
}

Python

def code(t, l, k):
	return 2.0 * (l * (l / (k * (k * (t * (k * k))))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(l * Float64(l / Float64(k * Float64(k * Float64(t * Float64(k * k)))))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (l * (l / (k * (k * (t * (k * k))))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(l * N[(l / N[(k * N[(k * N[(t * N[(k * k), $MachinePrecision]), $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

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2}{{k}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \frac{t}{{\ell}^{2}}} \]

   3. pow-sqr N/A

      \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot {k}^{2}\right)} \cdot \frac{t}{{\ell}^{2}}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

   10. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

   11. associate-*l* N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{{\ell}^{2}}} \]

   14. unpow2 N/A

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

   15. lower-*.f64 68.9

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

5. Simplified 68.9%

   \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \color{blue}{\left(k \cdot t\right)}}{\ell \cdot \ell}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{\ell \cdot \ell}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}{\ell \cdot \ell}}} \]

   8. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \left(\ell \cdot \ell\right) \]

   12. lower-*.f64 68.0

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \left(\ell \cdot \ell\right) \]

7. Applied egg-rr 68.0%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \cdot \left(\ell \cdot \ell\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \left(\ell \cdot \ell\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \left(\ell \cdot \ell\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   9. lower-/.f64 68.0

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot k\right)} \]

   12. *-commutative N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \left(k \cdot k\right)} \]

   13. associate-*l* N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   15. cube-mult N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{{k}^{3}}} \]

   16. lower-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot {k}^{3}}} \]

   17. cube-mult N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

   18. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   19. lower-*.f64 67.6

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

9. Applied egg-rr 67.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(k \cdot k\right)\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}} \]

    4. lift-*.f64 N/A

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

    5. associate-/l* N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}\right)} \]

    6. *-commutative N/A

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)} \cdot \ell\right)} \]

    7. lift-*.f64 N/A

       \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \cdot \ell\right) \]

    8. *-commutative N/A

       \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell\right) \]

    9. lift-*.f64 N/A

       \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell\right) \]

    10. associate-*r* N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(\left(k \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot t}} \cdot \ell\right) \]

    11. *-commutative N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)} \cdot t} \cdot \ell\right) \]

    12. lift-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right) \cdot t} \cdot \ell\right) \]

    13. associate-*l* N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \cdot \ell\right) \]

    14. lift-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right) \cdot t} \cdot \ell\right) \]

    15. lift-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \cdot \ell\right) \]

    16. lift-*.f64 N/A

        \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \cdot \ell\right) \]

    17. lower-*.f64 N/A

        \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t} \cdot \ell\right)} \]

11. Applied egg-rr 72.7%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \left(\left(t \cdot \left(k \cdot k\right)\right) \cdot k\right)} \cdot \ell\right)} \]

12. Final simplification 72.7%

    \[\leadsto 2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\right) \]
13. 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))))