Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 99.2%

Time: 15.3s

Alternatives: 12

Speedup: 9.6×

• 39.6% 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: 35.5% 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: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.0025:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot t\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot \frac{k\_m}{\ell}\right) \cdot t\right) \cdot \frac{\frac{k\_m}{\cos k\_m}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.0025)
     (/ 2.0 (* (* (* t_1 t) t_1) (fma 0.16666666666666666 (* k_m k_m) 1.0)))
     (/
      2.0
      (*
       (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) (/ k_m l)) t)
       (/ (/ k_m (cos k_m)) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.0025) {
		tmp = 2.0 / (((t_1 * t) * t_1) * fma(0.16666666666666666, (k_m * k_m), 1.0));
	} else {
		tmp = 2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * (k_m / l)) * t) * ((k_m / cos(k_m)) / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.0025)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * t) * t_1) * fma(0.16666666666666666, Float64(k_m * k_m), 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * Float64(k_m / l)) * t) * Float64(Float64(k_m / cos(k_m)) / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0025], N[(2.0 / N[(N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00250000000000000005

   1. Initial program 41.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{t}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{6} \cdot \color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{t}{{\ell}^{2}}\right)} \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{{k}^{2} \cdot \frac{1}{6}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{6} \cdot {k}^{2}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)} \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}} \]

      13. associate-/l* N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

   5. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\frac{\frac{{k}^{4}}{\ell}}{\ell} \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.4%

      \[\leadsto \frac{2}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \color{blue}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)}\right)} \]

   if 0.00250000000000000005 < k

   1. Initial program 24.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0025:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((((k_m / l) * pow(sin(k_m), 2.0)) * t) * ((k_m / cos(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 = 2.0d0 / ((((k_m / l) * (sin(k_m) ** 2.0d0)) * t) * ((k_m / cos(k_m)) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 94.5%

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Alternative 3: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 94.5%

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

    \[\leadsto \frac{2}{\frac{k}{\cos k \cdot \ell} \cdot \left(\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot t\right)} \]
11. Add Preprocessing

Alternative 4: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.0044:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot t\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right)}\\ \mathbf{elif}\;k\_m \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\cos \left(k\_m \cdot 2\right), -0.5, 0.5\right)}{\left(\cos k\_m \cdot \ell\right) \cdot \ell} \cdot \left(\left(k\_m \cdot k\_m\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\ell} \cdot \left(\left(\frac{k\_m}{\ell} \cdot {\sin k\_m}^{2}\right) \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.0044)
     (/ 2.0 (* (* (* t_1 t) t_1) (fma 0.16666666666666666 (* k_m k_m) 1.0)))
     (if (<= k_m 1.25e+150)
       (/
        2.0
        (*
         (/ (fma (cos (* k_m 2.0)) -0.5 0.5) (* (* (cos k_m) l) l))
         (* (* k_m k_m) t)))
       (/ 2.0 (* (/ k_m l) (* (* (/ k_m l) (pow (sin k_m) 2.0)) t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.0044) {
		tmp = 2.0 / (((t_1 * t) * t_1) * fma(0.16666666666666666, (k_m * k_m), 1.0));
	} else if (k_m <= 1.25e+150) {
		tmp = 2.0 / ((fma(cos((k_m * 2.0)), -0.5, 0.5) / ((cos(k_m) * l) * l)) * ((k_m * k_m) * t));
	} else {
		tmp = 2.0 / ((k_m / l) * (((k_m / l) * pow(sin(k_m), 2.0)) * t));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.0044)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * t) * t_1) * fma(0.16666666666666666, Float64(k_m * k_m), 1.0)));
	elseif (k_m <= 1.25e+150)
		tmp = Float64(2.0 / Float64(Float64(fma(cos(Float64(k_m * 2.0)), -0.5, 0.5) / Float64(Float64(cos(k_m) * l) * l)) * Float64(Float64(k_m * k_m) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / l) * Float64(Float64(Float64(k_m / l) * (sin(k_m) ^ 2.0)) * t)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0044], N[(2.0 / N[(N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.25e+150], N[(2.0 / N[(N[(N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(N[(N[(k$95$m / l), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.00440000000000000027

   1. Initial program 41.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{t}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{6} \cdot \color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{t}{{\ell}^{2}}\right)} \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{{k}^{2} \cdot \frac{1}{6}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{6} \cdot {k}^{2}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)} \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}} \]

      13. associate-/l* N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

   5. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\frac{\frac{{k}^{4}}{\ell}}{\ell} \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.4%

      \[\leadsto \frac{2}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \color{blue}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)}\right)} \]

   if 0.00440000000000000027 < k < 1.25000000000000002e150

   1. Initial program 12.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.1%

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 81.6%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 81.6%

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

   if 1.25000000000000002e150 < k

   1. Initial program 45.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 77.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0044:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\cos \left(k \cdot 2\right), -0.5, 0.5\right)}{\left(\cos k \cdot \ell\right) \cdot \ell} \cdot \left(\left(k \cdot k\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.0044:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot t\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(t, 0.5, \left(\cos \left(k\_m \cdot 2\right) \cdot -0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.0044)
     (/ 2.0 (* (* (* t_1 t) t_1) (fma 0.16666666666666666 (* k_m k_m) 1.0)))
     (/
      2.0
      (/
       (* (* (fma t 0.5 (* (* (cos (* k_m 2.0)) -0.5) t)) k_m) k_m)
       (* (* (cos k_m) l) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.0044) {
		tmp = 2.0 / (((t_1 * t) * t_1) * fma(0.16666666666666666, (k_m * k_m), 1.0));
	} else {
		tmp = 2.0 / (((fma(t, 0.5, ((cos((k_m * 2.0)) * -0.5) * t)) * k_m) * k_m) / ((cos(k_m) * l) * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.0044)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * t) * t_1) * fma(0.16666666666666666, Float64(k_m * k_m), 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(t, 0.5, Float64(Float64(cos(Float64(k_m * 2.0)) * -0.5) * t)) * k_m) * k_m) / Float64(Float64(cos(k_m) * l) * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0044], N[(2.0 / N[(N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * 0.5 + N[(N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00440000000000000027

   1. Initial program 41.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{t}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{6} \cdot \color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{t}{{\ell}^{2}}\right)} \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{{k}^{2} \cdot \frac{1}{6}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{6} \cdot {k}^{2}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)} \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}} \]

      13. associate-/l* N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

   5. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\frac{\frac{{k}^{4}}{\ell}}{\ell} \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.4%

      \[\leadsto \frac{2}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \color{blue}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)}\right)} \]

   if 0.00440000000000000027 < k

   1. Initial program 24.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 82.9%

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

   9. Applied rewrites 82.9%

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

   11. Applied rewrites 82.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0044:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(t, 0.5, \left(\cos \left(k \cdot 2\right) \cdot -0.5\right) \cdot t\right) \cdot k\right) \cdot k}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \mathbf{if}\;k\_m \leq 0.0044:\\ \;\;\;\;\frac{2}{\left(\left(t\_1 \cdot t\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(0.16666666666666666, k\_m \cdot k\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m)))
   (if (<= k_m 0.0044)
     (/ 2.0 (* (* (* t_1 t) t_1) (fma 0.16666666666666666 (* k_m k_m) 1.0)))
     (/
      2.0
      (/
       (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m)
       (* (* (cos k_m) l) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	double tmp;
	if (k_m <= 0.0044) {
		tmp = 2.0 / (((t_1 * t) * t_1) * fma(0.16666666666666666, (k_m * k_m), 1.0));
	} else {
		tmp = 2.0 / (((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m) / ((cos(k_m) * l) * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	tmp = 0.0
	if (k_m <= 0.0044)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * t) * t_1) * fma(0.16666666666666666, Float64(k_m * k_m), 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m) / Float64(Float64(cos(k_m) * l) * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0044], N[(2.0 / N[(N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.16666666666666666 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00440000000000000027

   1. Initial program 41.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}} + \frac{t}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{6} \cdot \color{blue}{\left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\right) \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \frac{t}{{\ell}^{2}}\right)} \cdot {k}^{4} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2}\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} + \frac{t}{{\ell}^{2}} \cdot {k}^{4}} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{6} \cdot {k}^{2} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{{k}^{2} \cdot \frac{1}{6}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot \frac{1}{6} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{6} \cdot {k}^{2}} + 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)} \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{k \cdot k}, 1\right) \cdot \left(\frac{t}{{\ell}^{2}} \cdot {k}^{4}\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}} \]

      13. associate-/l* N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(t \cdot \frac{{k}^{4}}{{\ell}^{2}}\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right) \cdot \color{blue}{\left(\frac{{k}^{4}}{{\ell}^{2}} \cdot t\right)}} \]

   5. Applied rewrites 75.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\frac{\frac{{k}^{4}}{\ell}}{\ell} \cdot t\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.4%

      \[\leadsto \frac{2}{\mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right) \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \color{blue}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right)}\right)} \]

   if 0.00440000000000000027 < k

   1. Initial program 24.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 82.9%

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

   9. Applied rewrites 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0044:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k\right) \cdot k}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 94.5%

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 78.3%

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

11. Final simplification 78.3%

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

Alternative 8: 76.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell} \cdot k\_m\\ \frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (/ k_m l) k_m))) (/ 2.0 (* (* t_1 t) t_1))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	return 2.0 / ((t_1 * t) * t_1);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    t_1 = (k_m / l) * k_m
    code = 2.0d0 / ((t_1 * t) * t_1)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (k_m / l) * k_m;
	return 2.0 / ((t_1 * t) * t_1);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m / l) * k_m
	return 2.0 / ((t_1 * t) * t_1)

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m / l) * k_m)
	return Float64(2.0 / Float64(Float64(t_1 * t) * t_1))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	t_1 = (k_m / l) * k_m;
	tmp = 2.0 / ((t_1 * t) * t_1);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m / l), $MachinePrecision] * k$95$m), $MachinePrecision]}, N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 71.2

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

5. Applied rewrites 71.2%

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

7. Applied rewrites 75.5%

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

9. Applied rewrites 76.6%

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

10. Final simplification 76.6%

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

Alternative 9: 73.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 71.2

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

5. Applied rewrites 71.2%

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

7. Applied rewrites 75.5%

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

9. Applied rewrites 76.3%

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

Alternative 10: 71.8% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 71.2

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

5. Applied rewrites 71.2%

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

7. Applied rewrites 75.5%

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

8. Final simplification 75.5%

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

Alternative 11: 76.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 71.2

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

5. Applied rewrites 71.2%

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

7. Applied rewrites 75.5%

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

9. Applied rewrites 76.3%

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

11. Applied rewrites 75.2%

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

12. Final simplification 75.2%

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

Alternative 12: 64.4% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.8%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 71.2

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

5. Applied rewrites 71.2%

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

7. Applied rewrites 75.5%

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

9. Applied rewrites 76.3%

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

11. Applied rewrites 68.0%

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

Reproduce

herbie shell --seed 2024250 
(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))))