Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 95.6%

Time: 17.3s

Alternatives: 17

Speedup: 11.0×

• 39.7% 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: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{k\_m}}{\frac{k\_m \cdot t}{\ell} \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.15e-8)
   (* (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5))) (/ l k_m))
   (/ (/ (* l 2.0) k_m) (* (/ (* k_m t) l) (* (tan k_m) (sin k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e-8) {
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = ((l * 2.0) / k_m) / (((k_m * t) / l) * (tan(k_m) * sin(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.15d-8) then
        tmp = (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0))) * (l / k_m)
    else
        tmp = ((l * 2.0d0) / k_m) / (((k_m * t) / l) * (tan(k_m) * sin(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.15e-8) {
		tmp = (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = ((l * 2.0) / k_m) / (((k_m * t) / l) * (Math.tan(k_m) * Math.sin(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.15e-8:
		tmp = (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m)
	else:
		tmp = ((l * 2.0) / k_m) / (((k_m * t) / l) * (math.tan(k_m) * math.sin(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.15e-8)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(l / k_m));
	else
		tmp = Float64(Float64(Float64(l * 2.0) / k_m) / Float64(Float64(Float64(k_m * t) / l) * Float64(tan(k_m) * sin(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.15e-8)
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	else
		tmp = ((l * 2.0) / k_m) / (((k_m * t) / l) * (tan(k_m) * sin(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.15e-8], N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(N[(N[(k$95$m * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1500000000000001e-8

   1. Initial program 36.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 75.0

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

   5. Applied rewrites 75.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.1%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 85.8

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

   12. Applied rewrites 85.8%

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

   if 2.1500000000000001e-8 < k

   1. Initial program 30.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.6%

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

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

      3. lift-*.f64 N/A

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

      4. lower-/.f64 95.6

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

   9. Applied rewrites 95.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{k}}{\frac{k \cdot t}{\ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell}{\tan k\_m} \cdot \frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ l (tan k_m)) (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / tan(k_m)) * (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l / tan(k_m)) * (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l / Math.tan(k_m)) * (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l / math.tan(k_m)) * (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / tan(k_m)) * Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l / tan(k_m)) * (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-sin.f64 75.4

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

5. Applied rewrites 75.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. associate-/r* N/A

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

7. Applied rewrites 95.2%

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

   1. associate-/r/ N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. times-frac N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. clear-num N/A

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

9. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)}} \]
10. Add Preprocessing

Alternative 3: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k\_m} \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6e-37)
   (* (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5))) (/ l k_m))
   (* (/ (* l 2.0) k_m) (/ l (* (* k_m t) (* (tan k_m) (sin k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6e-37) {
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = ((l * 2.0) / k_m) * (l / ((k_m * t) * (tan(k_m) * sin(k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 6d-37) then
        tmp = (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0))) * (l / k_m)
    else
        tmp = ((l * 2.0d0) / k_m) * (l / ((k_m * t) * (tan(k_m) * sin(k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6e-37) {
		tmp = (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = ((l * 2.0) / k_m) * (l / ((k_m * t) * (Math.tan(k_m) * Math.sin(k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 6e-37:
		tmp = (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m)
	else:
		tmp = ((l * 2.0) / k_m) * (l / ((k_m * t) * (math.tan(k_m) * math.sin(k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6e-37)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(l / k_m));
	else
		tmp = Float64(Float64(Float64(l * 2.0) / k_m) * Float64(l / Float64(Float64(k_m * t) * Float64(tan(k_m) * sin(k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 6e-37)
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	else
		tmp = ((l * 2.0) / k_m) * (l / ((k_m * t) * (tan(k_m) * sin(k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6e-37], N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * 2.0), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6e-37

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.0%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 85.5

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

   12. Applied rewrites 85.5%

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

   if 6e-37 < k

   1. Initial program 28.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-inv N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/r/ N/A

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

      14. associate-*l/ N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l/ N/A

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

   9. Applied rewrites 95.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00031:\\ \;\;\;\;\frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\mathsf{fma}\left(\ell, \left(k\_m \cdot k\_m\right) \cdot -0.3333333333333333, \ell\right)}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m} \cdot \left(\ell \cdot \frac{\ell}{\left(k\_m \cdot t\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.00031)
   (*
    (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5)))
    (/ (fma l (* (* k_m k_m) -0.3333333333333333) l) k_m))
   (* (/ 2.0 k_m) (* l (/ l (* (* k_m t) (* (tan k_m) (sin k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.00031) {
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (fma(l, ((k_m * k_m) * -0.3333333333333333), l) / k_m);
	} else {
		tmp = (2.0 / k_m) * (l * (l / ((k_m * t) * (tan(k_m) * sin(k_m)))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.00031)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(fma(l, Float64(Float64(k_m * k_m) * -0.3333333333333333), l) / k_m));
	else
		tmp = Float64(Float64(2.0 / k_m) * Float64(l * Float64(l / Float64(Float64(k_m * t) * Float64(tan(k_m) * sin(k_m))))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.00031], N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k$95$m), $MachinePrecision] * N[(l * N[(l / N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.1e-4

   1. Initial program 36.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.1%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\ell, \color{blue}{\left(k \cdot k\right)} \cdot \frac{-1}{3}, \ell\right)}{k} \cdot \frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \frac{1}{2}\right)} \]

       9. lower-*.f64 74.0

          \[\leadsto \frac{\mathsf{fma}\left(\ell, \color{blue}{\left(k \cdot k\right)} \cdot -0.3333333333333333, \ell\right)}{k} \cdot \frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \]

   12. Applied rewrites 74.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.3333333333333333, \ell\right)}{k}} \cdot \frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \]

   if 3.1e-4 < k

   1. Initial program 29.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 77.0

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

   5. Applied rewrites 77.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.5%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. div-inv N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 88.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00031:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.3333333333333333, \ell\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\ell \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k\_m \cdot \left(k\_m \cdot \left(\sin k\_m \cdot \left(\tan k\_m \cdot t\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.6e-37)
   (* (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5))) (/ l k_m))
   (* l (* l (/ 2.0 (* k_m (* k_m (* (sin k_m) (* (tan k_m) t)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.6e-37) {
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = l * (l * (2.0 / (k_m * (k_m * (sin(k_m) * (tan(k_m) * t))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.6d-37) then
        tmp = (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0))) * (l / k_m)
    else
        tmp = l * (l * (2.0d0 / (k_m * (k_m * (sin(k_m) * (tan(k_m) * t))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.6e-37) {
		tmp = (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = l * (l * (2.0 / (k_m * (k_m * (Math.sin(k_m) * (Math.tan(k_m) * t))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.6e-37:
		tmp = (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m)
	else:
		tmp = l * (l * (2.0 / (k_m * (k_m * (math.sin(k_m) * (math.tan(k_m) * t))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.6e-37)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(l / k_m));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(k_m * Float64(k_m * Float64(sin(k_m) * Float64(tan(k_m) * t)))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.6e-37)
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	else
		tmp = l * (l * (2.0 / (k_m * (k_m * (sin(k_m) * (tan(k_m) * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.6e-37], N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(k$95$m * N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.59999999999999936e-37

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.0%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 85.5

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

   12. Applied rewrites 85.5%

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

   if 8.59999999999999936e-37 < k

   1. Initial program 28.2%

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

   4. Applied rewrites 28.1%

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

   6. Applied rewrites 82.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 90.6

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

   8. Applied rewrites 90.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(\sin k \cdot \left(\tan k \cdot t\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k\_m \cdot \left(\sin k\_m \cdot \left(k\_m \cdot \left(k\_m \cdot t\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.6e-37)
   (* (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5))) (/ l k_m))
   (* l (* l (/ 2.0 (* (tan k_m) (* (sin k_m) (* k_m (* k_m t)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.6e-37) {
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = l * (l * (2.0 / (tan(k_m) * (sin(k_m) * (k_m * (k_m * t))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.6d-37) then
        tmp = (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0))) * (l / k_m)
    else
        tmp = l * (l * (2.0d0 / (tan(k_m) * (sin(k_m) * (k_m * (k_m * t))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.6e-37) {
		tmp = (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = l * (l * (2.0 / (Math.tan(k_m) * (Math.sin(k_m) * (k_m * (k_m * t))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.6e-37:
		tmp = (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m)
	else:
		tmp = l * (l * (2.0 / (math.tan(k_m) * (math.sin(k_m) * (k_m * (k_m * t))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.6e-37)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(l / k_m));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k_m) * Float64(sin(k_m) * Float64(k_m * Float64(k_m * t)))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.6e-37)
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	else
		tmp = l * (l * (2.0 / (tan(k_m) * (sin(k_m) * (k_m * (k_m * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.6e-37], N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.59999999999999936e-37

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.0%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 85.5

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

   12. Applied rewrites 85.5%

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

   if 8.59999999999999936e-37 < k

   1. Initial program 28.2%

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

   4. Applied rewrites 28.1%

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

   6. Applied rewrites 82.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 82.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 90.6

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

   8. Applied rewrites 90.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\ell}{k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\sin k\_m \cdot \left(t \cdot \left(k\_m \cdot \left(k\_m \cdot \tan k\_m\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 8.6e-37)
   (* (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5))) (/ l k_m))
   (* l (* 2.0 (/ l (* (sin k_m) (* t (* k_m (* k_m (tan k_m))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.6e-37) {
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = l * (2.0 * (l / (sin(k_m) * (t * (k_m * (k_m * tan(k_m)))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 8.6d-37) then
        tmp = (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0))) * (l / k_m)
    else
        tmp = l * (2.0d0 * (l / (sin(k_m) * (t * (k_m * (k_m * tan(k_m)))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 8.6e-37) {
		tmp = (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	} else {
		tmp = l * (2.0 * (l / (Math.sin(k_m) * (t * (k_m * (k_m * Math.tan(k_m)))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 8.6e-37:
		tmp = (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m)
	else:
		tmp = l * (2.0 * (l / (math.sin(k_m) * (t * (k_m * (k_m * math.tan(k_m)))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 8.6e-37)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(l / k_m));
	else
		tmp = Float64(l * Float64(2.0 * Float64(l / Float64(sin(k_m) * Float64(t * Float64(k_m * Float64(k_m * tan(k_m))))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 8.6e-37)
		tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
	else
		tmp = l * (2.0 * (l / (sin(k_m) * (t * (k_m * (k_m * tan(k_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 8.6e-37], N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 * N[(l / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t * N[(k$95$m * N[(k$95$m * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.59999999999999936e-37

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.9

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

   5. Applied rewrites 74.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l/ N/A

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

      10. associate-/r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. associate-/r* N/A

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

   7. Applied rewrites 95.0%

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

      1. associate-/r/ N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. times-frac N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. clear-num N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

      16. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 85.5

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

   12. Applied rewrites 85.5%

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

   if 8.59999999999999936e-37 < k

   1. Initial program 28.2%

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

   4. Applied rewrites 28.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 82.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(t \cdot \left(k \cdot \left(k \cdot \tan k\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{1}{\left(\left(k\_m \cdot t\right) \cdot \frac{\sin k\_m}{\ell}\right) \cdot \left(k\_m \cdot 0.5\right)} \cdot \frac{\ell}{k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ 1.0 (* (* (* k_m t) (/ (sin k_m) l)) (* k_m 0.5))) (/ l k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (1.0d0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5d0))) * (l / k_m)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (1.0 / (((k_m * t) * (Math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (1.0 / (((k_m * t) * (math.sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(1.0 / Float64(Float64(Float64(k_m * t) * Float64(sin(k_m) / l)) * Float64(k_m * 0.5))) * Float64(l / k_m))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (1.0 / (((k_m * t) * (sin(k_m) / l)) * (k_m * 0.5))) * (l / k_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(1.0 / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-sin.f64 75.4

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

5. Applied rewrites 75.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. associate-/r* N/A

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

7. Applied rewrites 95.2%

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

   1. associate-/r/ N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. times-frac N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. clear-num N/A

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{1}{\frac{\frac{k \cdot t}{\ell} \cdot \sin k}{\frac{2}{k}}}} \]

9. Applied rewrites 97.1%

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

10. Taylor expanded in k around 0

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

    1. lower-/.f64 77.1

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

12. Applied rewrites 77.1%

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

13. Final simplification 77.1%

    \[\leadsto \frac{1}{\left(\left(k \cdot t\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot 0.5\right)} \cdot \frac{\ell}{k} \]
14. Add Preprocessing

Alternative 9: 75.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{2}{\frac{k\_m}{\ell}}}{\left(\left(k\_m \cdot t\right) \cdot \frac{k\_m}{\ell}\right) \cdot \mathsf{fma}\left(0.16666666666666666, k\_m \cdot \left(k\_m \cdot k\_m\right), k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  (/ 2.0 (/ k_m l))
  (*
   (* (* k_m t) (/ k_m l))
   (fma 0.16666666666666666 (* k_m (* k_m k_m)) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (2.0 / (k_m / l)) / (((k_m * t) * (k_m / l)) * fma(0.16666666666666666, (k_m * (k_m * k_m)), k_m));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(2.0 / Float64(k_m / l)) / Float64(Float64(Float64(k_m * t) * Float64(k_m / l)) * fma(0.16666666666666666, Float64(k_m * Float64(k_m * k_m)), k_m)))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(2.0 / N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-sin.f64 75.4

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

5. Applied rewrites 75.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. associate-/r* N/A

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

7. Applied rewrites 95.2%

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

8. Taylor expanded in k around 0

   \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{{k}^{3} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)}} \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}\right) \cdot {k}^{3} + \frac{t}{\ell} \cdot {k}^{3}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{1}{6}\right)} \cdot {k}^{3} + \frac{t}{\ell} \cdot {k}^{3}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right)} + \frac{t}{\ell} \cdot {k}^{3}} \]

   4. unpow3 N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \frac{t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot k\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \frac{t}{\ell} \cdot \left(\color{blue}{{k}^{2}} \cdot k\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \color{blue}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot k}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot k} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot k} \]

   9. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{3}, k\right)}} \]

   17. cube-mult N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot \left(k \cdot k\right)}, k\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \color{blue}{{k}^{2}}, k\right)} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot {k}^{2}}, k\right)} \]

   20. unpow2 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \color{blue}{\left(k \cdot k\right)}, k\right)} \]

   21. lower-*.f64 73.8

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot \color{blue}{\left(k \cdot k\right)}, k\right)} \]

10. Applied rewrites 73.8%

    \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot \left(k \cdot k\right), k\right)}} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \left(k \cdot k\right), k\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{\left(k \cdot t\right)} \cdot k}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \left(k \cdot k\right), k\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{\left(k \cdot t\right)} \cdot k}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \left(k \cdot k\right), k\right)} \]

    4. associate-/l* N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\left(k \cdot t\right) \cdot \frac{k}{\ell}\right)} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \left(k \cdot k\right), k\right)} \]

    5. lift-/.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\left(\left(k \cdot t\right) \cdot \color{blue}{\frac{k}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \left(k \cdot k\right), k\right)} \]

    6. lower-*.f64 76.0

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\left(k \cdot t\right) \cdot \frac{k}{\ell}\right)} \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot \left(k \cdot k\right), k\right)} \]

12. Applied rewrites 76.0%

    \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\left(k \cdot t\right) \cdot \frac{k}{\ell}\right)} \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot \left(k \cdot k\right), k\right)} \]
13. Add Preprocessing

Alternative 10: 74.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{\frac{k\_m}{\ell \cdot 2} \cdot \mathsf{fma}\left(k\_m, \left(k\_m \cdot k\_m\right) \cdot 0.16666666666666666, k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  (/ l (* k_m (* k_m t)))
  (* (/ k_m (* l 2.0)) (fma k_m (* (* k_m k_m) 0.16666666666666666) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l / (k_m * (k_m * t))) / ((k_m / (l * 2.0)) * fma(k_m, ((k_m * k_m) * 0.16666666666666666), k_m));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l / Float64(k_m * Float64(k_m * t))) / Float64(Float64(k_m / Float64(l * 2.0)) * fma(k_m, Float64(Float64(k_m * k_m) * 0.16666666666666666), k_m)))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m / N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-sin.f64 75.4

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

5. Applied rewrites 75.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l/ N/A

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

   10. associate-/r/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. associate-/r* N/A

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

7. Applied rewrites 95.2%

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

8. Taylor expanded in k around 0

   \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{{k}^{3} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell} + \frac{t}{\ell}\right)}} \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{\ell}\right) \cdot {k}^{3} + \frac{t}{\ell} \cdot {k}^{3}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{1}{6}\right)} \cdot {k}^{3} + \frac{t}{\ell} \cdot {k}^{3}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right)} + \frac{t}{\ell} \cdot {k}^{3}} \]

   4. unpow3 N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \frac{t}{\ell} \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot k\right)}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \frac{t}{\ell} \cdot \left(\color{blue}{{k}^{2}} \cdot k\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \color{blue}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot k}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \color{blue}{\left({k}^{2} \cdot \frac{t}{\ell}\right)} \cdot k} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3}\right) + \color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot k} \]

   9. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\ell} \cdot \left(\frac{1}{6} \cdot {k}^{3} + k\right)} \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{3}, k\right)}} \]

   17. cube-mult N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot \left(k \cdot k\right)}, k\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \color{blue}{{k}^{2}}, k\right)} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot {k}^{2}}, k\right)} \]

   20. unpow2 N/A

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot \color{blue}{\left(k \cdot k\right)}, k\right)} \]

   21. lower-*.f64 73.8

       \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot \color{blue}{\left(k \cdot k\right)}, k\right)} \]

10. Applied rewrites 73.8%

    \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell} \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot \left(k \cdot k\right), k\right)}} \]

12. Applied rewrites 75.4%

    \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{\frac{k}{\ell \cdot 2} \cdot \mathsf{fma}\left(k, \left(k \cdot k\right) \cdot 0.16666666666666666, k\right)}} \]
13. Add Preprocessing

Alternative 11: 72.9% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.5%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 60.3

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

5. Applied rewrites 60.3%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 68.0

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 69.0

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in k around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 69.4

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

10. Applied rewrites 69.4%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*r/ N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. associate-*r* N/A

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

    10. lift-*.f64 N/A

        \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \]

    11. times-frac N/A

        \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}} \]

    12. lower-*.f64 N/A

        \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}} \]

    13. lower-/.f64 N/A

        \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k}} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)} \]

    14. lift-*.f64 N/A

        \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)} \]

    15. *-commutative N/A

        \[\leadsto \frac{\color{blue}{\ell \cdot 2}}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)} \]

    16. lower-*.f64 N/A

        \[\leadsto \frac{\color{blue}{\ell \cdot 2}}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)} \]

    17. lower-/.f64 73.9

        \[\leadsto \frac{\ell \cdot 2}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot \left(k \cdot k\right)}} \]

    18. lift-*.f64 N/A

        \[\leadsto \frac{\ell \cdot 2}{k \cdot k} \cdot \frac{\ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

    19. *-commutative N/A

        \[\leadsto \frac{\ell \cdot 2}{k \cdot k} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

    20. lift-*.f64 N/A

        \[\leadsto \frac{\ell \cdot 2}{k \cdot k} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    21. associate-*l* N/A

        \[\leadsto \frac{\ell \cdot 2}{k \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

    22. lift-*.f64 N/A

        \[\leadsto \frac{\ell \cdot 2}{k \cdot k} \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot t\right)}} \]

12. Applied rewrites 73.9%

    \[\leadsto \color{blue}{\frac{\ell \cdot 2}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}} \]

13. Final simplification 73.9%

    \[\leadsto \frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot 2}{k \cdot k} \]
14. Add Preprocessing

Alternative 12: 72.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot 2}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* l 2.0) (* t (* k_m k_m))) (/ l (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * 2.0) / (t * (k_m * k_m))) * (l / (k_m * k_m));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * 2.0d0) / (t * (k_m * k_m))) * (l / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * 2.0) / (t * (k_m * k_m))) * (l / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * 2.0) / (t * (k_m * k_m))) * (l / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * 2.0) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * 2.0) / (t * (k_m * k_m))) * (l / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * 2.0), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. lower-*.f64 60.3

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}} \]

   5. times-frac N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot \ell}}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{k \cdot k} \]

   10. lower-/.f64 73.9

       \[\leadsto \frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot k}} \]

7. Applied rewrites 73.9%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}} \]

8. Final simplification 73.9%

   \[\leadsto \frac{\ell \cdot 2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k} \]
9. Add Preprocessing

Alternative 13: 72.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k\_m \cdot \left(k\_m \cdot t\right)}}{k\_m \cdot k\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* l 2.0) (/ (/ l (* k_m (* k_m t))) (* k_m k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * 2.0) * ((l / (k_m * (k_m * t))) / (k_m * k_m));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * ((l / (k_m * (k_m * t))) / (k_m * k_m))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * 2.0) * ((l / (k_m * (k_m * t))) / (k_m * k_m));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * 2.0) * ((l / (k_m * (k_m * t))) / (k_m * k_m))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * 2.0) * Float64(Float64(l / Float64(k_m * Float64(k_m * t))) / Float64(k_m * k_m)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * 2.0) * ((l / (k_m * (k_m * t))) / (k_m * k_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. lower-*.f64 60.3

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   9. lower-/.f64 68.0

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   14. associate-*r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)} \cdot t} \]

   15. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right)} \cdot t} \]

   16. associate-*l* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   18. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   19. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   20. lower-*.f64 69.0

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

7. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left({k}^{3} \cdot t\right)}} \]
9. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \color{blue}{{k}^{2}}\right) \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot t\right)\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot t\right)\right)}} \]

   5. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \]

   7. unpow2 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

   8. lower-*.f64 69.4

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

10. Applied rewrites 69.4%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

    4. associate-/r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

    5. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

    6. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

    7. associate-*r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot k\right)}} \]

    8. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot k\right)} \]

    9. associate-/r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\frac{\ell}{k}}{k \cdot t}}{k \cdot k}} \]

    10. associate-/l/ N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot k}}}{k \cdot k} \]

    11. lift-*.f64 N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot k}}{k \cdot k} \]

    12. *-commutative N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{\left(t \cdot k\right)} \cdot k}}{k \cdot k} \]

    13. associate-*r* N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{t \cdot \left(k \cdot k\right)}}}{k \cdot k} \]

    14. lift-*.f64 N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{t \cdot \color{blue}{\left(k \cdot k\right)}}}{k \cdot k} \]

    15. lift-*.f64 N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{t \cdot \left(k \cdot k\right)}}}{k \cdot k} \]

    16. lower-/.f64 N/A

        \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k}} \]

12. Applied rewrites 72.0%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k}} \]

13. Final simplification 72.0%

    \[\leadsto \left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k} \]
14. Add Preprocessing

Alternative 14: 70.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k\_m \cdot t}}{k\_m \cdot \left(k\_m \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* l 2.0) (/ (/ l (* k_m t)) (* k_m (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * 2.0) * ((l / (k_m * t)) / (k_m * (k_m * k_m)));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * ((l / (k_m * t)) / (k_m * (k_m * k_m)))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * 2.0) * ((l / (k_m * t)) / (k_m * (k_m * k_m)));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * 2.0) * ((l / (k_m * t)) / (k_m * (k_m * k_m)))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * 2.0) * Float64(Float64(l / Float64(k_m * t)) / Float64(k_m * Float64(k_m * k_m))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * 2.0) * ((l / (k_m * t)) / (k_m * (k_m * k_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. lower-*.f64 60.3

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   9. lower-/.f64 68.0

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   14. associate-*r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)} \cdot t} \]

   15. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right)} \cdot t} \]

   16. associate-*l* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   18. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   19. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   20. lower-*.f64 69.0

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

7. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   5. clear-num N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{1}{\frac{k}{\ell}}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{1}{\color{blue}{\frac{k}{\ell}}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   8. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   9. associate-/r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\frac{1}{\frac{k}{\ell}}}{t}}{k \cdot \left(k \cdot k\right)}} \]

   10. lift-/.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\frac{1}{\color{blue}{\frac{k}{\ell}}}}{t}}{k \cdot \left(k \cdot k\right)} \]

   11. clear-num N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\color{blue}{\frac{\ell}{k}}}{t}}{k \cdot \left(k \cdot k\right)} \]

   12. associate-/r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{k \cdot \left(k \cdot k\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{\color{blue}{k \cdot t}}}{k \cdot \left(k \cdot k\right)} \]

   14. lower-/.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(k \cdot k\right)}} \]

   15. lower-/.f64 70.1

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{k \cdot \left(k \cdot k\right)} \]

9. Applied rewrites 70.1%

   \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(k \cdot k\right)}} \]

10. Final simplification 70.1%

    \[\leadsto \left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k \cdot t}}{k \cdot \left(k \cdot k\right)} \]
11. Add Preprocessing

Alternative 15: 70.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k\_m}}{t \cdot \left(k\_m \cdot \left(k\_m \cdot k\_m\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* l 2.0) (/ (/ l k_m) (* t (* k_m (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * 2.0) * ((l / k_m) / (t * (k_m * (k_m * k_m))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * ((l / k_m) / (t * (k_m * (k_m * k_m))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * 2.0) * ((l / k_m) / (t * (k_m * (k_m * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * 2.0) * ((l / k_m) / (t * (k_m * (k_m * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * 2.0) * Float64(Float64(l / k_m) / Float64(t * Float64(k_m * Float64(k_m * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * 2.0) * ((l / k_m) / (t * (k_m * (k_m * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[(t * N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. lower-*.f64 60.3

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   9. lower-/.f64 68.0

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   14. associate-*r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)} \cdot t} \]

   15. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right)} \cdot t} \]

   16. associate-*l* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   18. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   19. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   20. lower-*.f64 69.0

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

7. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   5. clear-num N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{1}{\frac{k}{\ell}}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   6. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{1}{\color{blue}{\frac{k}{\ell}}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   7. lower-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   8. lift-/.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{1}{\color{blue}{\frac{k}{\ell}}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. clear-num N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{\ell}{k}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   10. lower-/.f64 69.8

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\color{blue}{\frac{\ell}{k}}}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   12. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

   13. lower-*.f64 69.8

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\frac{\ell}{k}}{\color{blue}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

9. Applied rewrites 69.8%

   \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)}} \]

10. Final simplification 69.8%

    \[\leadsto \left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k}}{t \cdot \left(k \cdot \left(k \cdot k\right)\right)} \]
11. Add Preprocessing

Alternative 16: 70.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\ell \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (* l (/ 2.0 (* (* k_m k_m) (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l * (2.0 / ((k_m * k_m) * (t * (k_m * k_m)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = l * (l * (2.0d0 / ((k_m * k_m) * (t * (k_m * k_m)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l * (2.0 / ((k_m * k_m) * (t * (k_m * k_m)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l * (2.0 / ((k_m * k_m) * (t * (k_m * k_m)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l * Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(t * Float64(k_m * k_m))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l * (2.0 / ((k_m * k_m) * (t * (k_m * k_m)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

4. Applied rewrites 26.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot k}{t \cdot t} \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell \cdot \ell}}} \]

6. Applied rewrites 85.2%

   \[\leadsto \color{blue}{\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell} \]

7. Taylor expanded in k around 0

   \[\leadsto \left(\frac{2}{\color{blue}{{k}^{2}} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell \]

   2. lower-*.f64 69.4

      \[\leadsto \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell \]

9. Applied rewrites 69.4%

   \[\leadsto \left(\frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell \]

10. Final simplification 69.4%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right) \]
11. Add Preprocessing

Alternative 17: 70.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot 2\right) \cdot \frac{\ell}{k\_m \cdot \left(k\_m \cdot \left(t \cdot \left(k\_m \cdot k\_m\right)\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (* l 2.0) (/ l (* k_m (* k_m (* t (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * 2.0) * (l / (k_m * (k_m * (t * (k_m * k_m)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * 2.0d0) * (l / (k_m * (k_m * (t * (k_m * k_m)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * 2.0) * (l / (k_m * (k_m * (t * (k_m * k_m)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * 2.0) * (l / (k_m * (k_m * (t * (k_m * k_m)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * 2.0) * Float64(l / Float64(k_m * Float64(k_m * Float64(t * Float64(k_m * k_m))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * 2.0) * (l / (k_m * (k_m * (t * (k_m * k_m)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(k$95$m * N[(k$95$m * N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot {\ell}^{2}}}{{k}^{4} \cdot t} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]

   9. pow-sqr N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \color{blue}{\left({k}^{2} \cdot {k}^{2}\right)}} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   14. lower-*.f64 60.3

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Applied rewrites 60.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{t \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot \frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \]

   9. lower-/.f64 68.0

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{\ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}} \]

   11. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot t} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot t} \]

   14. associate-*r* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot k\right)} \cdot t} \]

   15. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot k\right)\right)} \cdot t} \]

   16. associate-*l* N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{k \cdot \left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   18. lower-*.f64 N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(k \cdot k\right) \cdot k\right) \cdot t\right)}} \]

   19. *-commutative N/A

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   20. lower-*.f64 69.0

       \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

7. Applied rewrites 69.0%

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \left(k \cdot k\right)\right) \cdot t\right)}} \]

8. Taylor expanded in k around 0

   \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left({k}^{3} \cdot t\right)}} \]
9. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot k\right)\right)} \cdot t\right)} \]

   2. unpow2 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot \color{blue}{{k}^{2}}\right) \cdot t\right)} \]

   3. associate-*l* N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot t\right)\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left({k}^{2} \cdot t\right)\right)}} \]

   5. *-commutative N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \]

   7. unpow2 N/A

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

   8. lower-*.f64 69.4

      \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

10. Applied rewrites 69.4%

    \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]

11. Final simplification 69.4%

    \[\leadsto \left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(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))))