Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 92.5%

Time: 16.9s

Alternatives: 13

Speedup: 13.2×

• 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: 35.4% 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: 92.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \left(k\_m \cdot k\_m\right)} \cdot \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell + \ell}{\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot \left(k\_m \cdot t\right)} \cdot \frac{\cos k\_m}{k\_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.2e-5)
   (* (/ (+ l l) (* t (* k_m k_m))) (/ l (* k_m k_m)))
   (*
    l
    (*
     (/ (+ l l) (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) (* k_m t)))
     (/ (cos k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-5) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = l * (((l + l) / ((0.5 - (0.5 * cos((k_m + k_m)))) * (k_m * t))) * (cos(k_m) / k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9.2d-5) then
        tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
    else
        tmp = l * (((l + l) / ((0.5d0 - (0.5d0 * cos((k_m + k_m)))) * (k_m * t))) * (cos(k_m) / k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.2e-5) {
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	} else {
		tmp = l * (((l + l) / ((0.5 - (0.5 * Math.cos((k_m + k_m)))) * (k_m * t))) * (Math.cos(k_m) / k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9.2e-5:
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m))
	else:
		tmp = l * (((l + l) / ((0.5 - (0.5 * math.cos((k_m + k_m)))) * (k_m * t))) * (math.cos(k_m) / k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.2e-5)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * Float64(k_m * k_m))) * Float64(l / Float64(k_m * k_m)));
	else
		tmp = Float64(l * Float64(Float64(Float64(l + l) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * Float64(k_m * t))) * Float64(cos(k_m) / k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9.2e-5)
		tmp = ((l + l) / (t * (k_m * k_m))) * (l / (k_m * k_m));
	else
		tmp = l * (((l + l) / ((0.5 - (0.5 * cos((k_m + k_m)))) * (k_m * t))) * (cos(k_m) / k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.2e-5], N[(N[(N[(l + l), $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], N[(l * N[(N[(N[(l + l), $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.20000000000000001e-5

   1. Initial program 33.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 77.9%

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

   if 9.20000000000000001e-5 < k

   1. Initial program 21.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 18.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 21.7%

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

   7. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 82.2

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

   9. Applied rewrites 82.2%

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

   11. Applied rewrites 91.4%

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

Alternative 2: 88.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{+145}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\ell}{t \cdot \sin k\_m}}{k\_m \cdot \left(k\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(2 \cdot \ell\right) \cdot \frac{\cos k\_m}{k\_m \cdot \left(k\_m \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e+145)
   (* l (/ (* 2.0 (/ l (* t (sin k_m)))) (* k_m (* k_m (tan k_m)))))
   (*
    l
    (*
     (* 2.0 l)
     (/ (cos k_m) (* k_m (* k_m (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e+145) {
		tmp = l * ((2.0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))));
	} else {
		tmp = l * ((2.0 * l) * (cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * cos((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 <= 1.7d+145) then
        tmp = l * ((2.0d0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))))
    else
        tmp = l * ((2.0d0 * l) * (cos(k_m) / (k_m * (k_m * ((0.5d0 - (0.5d0 * cos((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 <= 1.7e+145) {
		tmp = l * ((2.0 * (l / (t * Math.sin(k_m)))) / (k_m * (k_m * Math.tan(k_m))));
	} else {
		tmp = l * ((2.0 * l) * (Math.cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * Math.cos((k_m + k_m)))) * t)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e+145:
		tmp = l * ((2.0 * (l / (t * math.sin(k_m)))) / (k_m * (k_m * math.tan(k_m))))
	else:
		tmp = l * ((2.0 * l) * (math.cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * math.cos((k_m + k_m)))) * t)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e+145)
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l / Float64(t * sin(k_m)))) / Float64(k_m * Float64(k_m * tan(k_m)))));
	else
		tmp = Float64(l * Float64(Float64(2.0 * l) * Float64(cos(k_m) / Float64(k_m * Float64(k_m * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e+145)
		tmp = l * ((2.0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))));
	else
		tmp = l * ((2.0 * l) * (cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * cos((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, 1.7e+145], N[(l * N[(N[(2.0 * N[(l / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(2.0 * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e145

   1. Initial program 30.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 25.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 84.6

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

   11. Applied rewrites 84.6%

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

   if 1.7e145 < k

   1. Initial program 29.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 23.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 24.5%

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

   7. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 71.8

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

   9. Applied rewrites 71.8%

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

   11. Applied rewrites 71.8%

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

Alternative 3: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{+145}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\ell}{t \cdot \sin k\_m}}{k\_m \cdot \left(k\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \left(\ell + \ell\right)\right) \cdot \frac{\cos k\_m}{k\_m \cdot \left(t \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e+145)
   (* l (/ (* 2.0 (/ l (* t (sin k_m)))) (* k_m (* k_m (tan k_m)))))
   (*
    (* l (+ l l))
    (/ (cos k_m) (* k_m (* t (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e+145) {
		tmp = l * ((2.0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))));
	} else {
		tmp = (l * (l + l)) * (cos(k_m) / (k_m * (t * ((0.5 - (0.5 * cos((k_m + k_m)))) * k_m))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.7d+145) then
        tmp = l * ((2.0d0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))))
    else
        tmp = (l * (l + l)) * (cos(k_m) / (k_m * (t * ((0.5d0 - (0.5d0 * cos((k_m + k_m)))) * k_m))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e+145) {
		tmp = l * ((2.0 * (l / (t * Math.sin(k_m)))) / (k_m * (k_m * Math.tan(k_m))));
	} else {
		tmp = (l * (l + l)) * (Math.cos(k_m) / (k_m * (t * ((0.5 - (0.5 * Math.cos((k_m + k_m)))) * k_m))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e+145:
		tmp = l * ((2.0 * (l / (t * math.sin(k_m)))) / (k_m * (k_m * math.tan(k_m))))
	else:
		tmp = (l * (l + l)) * (math.cos(k_m) / (k_m * (t * ((0.5 - (0.5 * math.cos((k_m + k_m)))) * k_m))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e+145)
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l / Float64(t * sin(k_m)))) / Float64(k_m * Float64(k_m * tan(k_m)))));
	else
		tmp = Float64(Float64(l * Float64(l + l)) * Float64(cos(k_m) / Float64(k_m * Float64(t * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * k_m)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e+145)
		tmp = l * ((2.0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))));
	else
		tmp = (l * (l + l)) * (cos(k_m) / (k_m * (t * ((0.5 - (0.5 * cos((k_m + k_m)))) * k_m))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.7e+145], N[(l * N[(N[(2.0 * N[(l / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(k$95$m * N[(t * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e145

   1. Initial program 30.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 25.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 84.6

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

   11. Applied rewrites 84.6%

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

   if 1.7e145 < k

   1. Initial program 29.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. count-2 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-sin.f64 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 59.9%

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

Alternative 4: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.7 \cdot 10^{+145}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \frac{\ell}{t \cdot \sin k\_m}}{k\_m \cdot \left(k\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \left(\ell + \ell\right)\right) \cdot \frac{\cos k\_m}{k\_m \cdot \left(k\_m \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e+145)
   (* l (/ (* 2.0 (/ l (* t (sin k_m)))) (* k_m (* k_m (tan k_m)))))
   (*
    (* l (+ l l))
    (/ (cos k_m) (* k_m (* k_m (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e+145) {
		tmp = l * ((2.0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))));
	} else {
		tmp = (l * (l + l)) * (cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * cos((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 <= 1.7d+145) then
        tmp = l * ((2.0d0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))))
    else
        tmp = (l * (l + l)) * (cos(k_m) / (k_m * (k_m * ((0.5d0 - (0.5d0 * cos((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 <= 1.7e+145) {
		tmp = l * ((2.0 * (l / (t * Math.sin(k_m)))) / (k_m * (k_m * Math.tan(k_m))));
	} else {
		tmp = (l * (l + l)) * (Math.cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * Math.cos((k_m + k_m)))) * t))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.7e+145:
		tmp = l * ((2.0 * (l / (t * math.sin(k_m)))) / (k_m * (k_m * math.tan(k_m))))
	else:
		tmp = (l * (l + l)) * (math.cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * math.cos((k_m + k_m)))) * t))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e+145)
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l / Float64(t * sin(k_m)))) / Float64(k_m * Float64(k_m * tan(k_m)))));
	else
		tmp = Float64(Float64(l * Float64(l + l)) * Float64(cos(k_m) / Float64(k_m * Float64(k_m * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t)))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.7e+145)
		tmp = l * ((2.0 * (l / (t * sin(k_m)))) / (k_m * (k_m * tan(k_m))));
	else
		tmp = (l * (l + l)) * (cos(k_m) / (k_m * (k_m * ((0.5 - (0.5 * cos((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, 1.7e+145], N[(l * N[(N[(2.0 * N[(l / N[(t * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e145

   1. Initial program 30.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/r* N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 25.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 34.6

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

   8. Applied rewrites 34.6%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 84.6

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

   11. Applied rewrites 84.6%

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

   if 1.7e145 < k

   1. Initial program 29.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. count-2 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-sin.f64 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 59.9%

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

Alternative 5: 84.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

4. Applied rewrites 25.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 29.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 33.2

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

8. Applied rewrites 33.2%

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

9. Taylor expanded in l around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sin.f64 81.0

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

11. Applied rewrites 81.0%

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

Alternative 6: 53.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.76 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{t \cdot k\_m}}{k\_m \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \frac{\frac{\ell}{t}}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.76e-82)
   (/ (/ (+ l l) (* t k_m)) (* k_m (* k_m k_m)))
   (* (+ l l) (/ (/ l t) (* (* k_m k_m) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.76e-82) {
		tmp = ((l + l) / (t * k_m)) / (k_m * (k_m * k_m));
	} else {
		tmp = (l + l) * ((l / t) / ((k_m * k_m) * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.76d-82) then
        tmp = ((l + l) / (t * k_m)) / (k_m * (k_m * k_m))
    else
        tmp = (l + l) * ((l / t) / ((k_m * k_m) * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.76e-82) {
		tmp = ((l + l) / (t * k_m)) / (k_m * (k_m * k_m));
	} else {
		tmp = (l + l) * ((l / t) / ((k_m * k_m) * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.76e-82:
		tmp = ((l + l) / (t * k_m)) / (k_m * (k_m * k_m))
	else:
		tmp = (l + l) * ((l / t) / ((k_m * k_m) * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.76e-82)
		tmp = Float64(Float64(Float64(l + l) / Float64(t * k_m)) / Float64(k_m * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(l + l) * Float64(Float64(l / t) / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.76e-82)
		tmp = ((l + l) / (t * k_m)) / (k_m * (k_m * k_m));
	else
		tmp = (l + l) * ((l / t) / ((k_m * k_m) * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.76e-82], N[(N[(N[(l + l), $MachinePrecision] / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.76000000000000006e-82

   1. Initial program 33.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 34.1%

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

   9. Applied rewrites 34.7%

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

   if 1.76000000000000006e-82 < k

   1. Initial program 24.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 41.9%

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

   9. Applied rewrites 55.8%

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

Alternative 7: 72.9% accurate, 10.0× speedup

Math

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

FPCore

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

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l + l) / (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 + l) / 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 + l) / (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 + l), $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 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 70.3%

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

Alternative 8: 68.4% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 66.5%

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

Alternative 9: 68.4% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 66.5%

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

Alternative 10: 68.4% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 36.6%

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

9. Applied rewrites 66.5%

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

Alternative 11: 42.5% accurate, 13.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 36.6%

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

9. Applied rewrites 36.7%

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

Alternative 12: 42.5% accurate, 13.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 36.6%

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

9. Applied rewrites 36.7%

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

Alternative 13: 42.6% accurate, 13.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

3. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. count-2 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. pow-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 61.8

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

5. Applied rewrites 61.8%

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

7. Applied rewrites 36.6%

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

Reproduce

herbie shell --seed 2024254 
(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))))