Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.8% → 96.8%

Time: 11.9s

Alternatives: 12

Speedup: 11.0×

• 39.1% 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: 34.8% 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: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell \cdot \cos k\_m}\\ \mathbf{if}\;k\_m \leq 8 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, -0.3333333333333333, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell} \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left({\sin k\_m}^{2} \cdot \frac{k\_m}{\ell}\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* l (cos k_m)))))
   (if (<= k_m 8e-141)
     (/
      2.0
      (*
       (/
        (* (* (* (fma (* (* k_m k_m) t) -0.3333333333333333 t) k_m) k_m) k_m)
        l)
       t_1))
     (/ 2.0 (* (* t (* (pow (sin k_m) 2.0) (/ k_m l))) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (l * cos(k_m));
	double tmp;
	if (k_m <= 8e-141) {
		tmp = 2.0 / (((((fma(((k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * t_1);
	} else {
		tmp = 2.0 / ((t * (pow(sin(k_m), 2.0) * (k_m / l))) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(l * cos(k_m)))
	tmp = 0.0
	if (k_m <= 8e-141)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64((sin(k_m) ^ 2.0) * Float64(k_m / l))) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8e-141], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000003e-141

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 76.7%

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

   if 8.0000000000000003e-141 < k

   1. Initial program 26.8%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 98.7%

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

Alternative 2: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / (((((k_m / l) * t) * sin(k_m)) * sin(k_m)) * (k_m / (l * cos(k_m))));
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] * t), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.6

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

5. Applied rewrites 91.6%

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

7. Applied rewrites 96.3%

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

9. Applied rewrites 97.3%

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

Alternative 3: 96.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.6

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

5. Applied rewrites 91.6%

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

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.7%

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

Alternative 4: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell \cdot \cos k\_m}\\ \mathbf{if}\;k\_m \leq 0.0026:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, -0.3333333333333333, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell} \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right) \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* l (cos k_m)))))
   (if (<= k_m 0.0026)
     (/
      2.0
      (*
       (/
        (* (* (* (fma (* (* k_m k_m) t) -0.3333333333333333 t) k_m) k_m) k_m)
        l)
       t_1))
     (/ 2.0 (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) (* t (/ k_m l))) t_1)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (l * cos(k_m));
	double tmp;
	if (k_m <= 0.0026) {
		tmp = 2.0 / (((((fma(((k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * t_1);
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((k_m + k_m)))) * (t * (k_m / l))) * t_1);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(l * cos(k_m)))
	tmp = 0.0
	if (k_m <= 0.0026)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * Float64(t * Float64(k_m / l))) * t_1));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 0.0026], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0025999999999999999

   1. Initial program 35.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 78.9%

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

   if 0.0025999999999999999 < k

   1. Initial program 28.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 98.9%

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

Alternative 5: 85.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.025:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, -0.3333333333333333, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell} \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right) \cdot k\_m\right) \cdot \frac{k\_m}{\left(\cos k\_m \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.025)
   (/
    2.0
    (*
     (/
      (* (* (* (fma (* (* k_m k_m) t) -0.3333333333333333 t) k_m) k_m) k_m)
      l)
     (/ k_m (* l (cos k_m)))))
   (/
    2.0
    (*
     (* (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) 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 tmp;
	if (k_m <= 0.025) {
		tmp = 2.0 / (((((fma(((k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * (k_m / (l * cos(k_m))));
	} else {
		tmp = 2.0 / ((((0.5 - (0.5 * cos((k_m + k_m)))) * t) * k_m) * (k_m / ((cos(k_m) * l) * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.025)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * Float64(k_m / Float64(l * cos(k_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t) * k_m) * Float64(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_] := If[LessEqual[k$95$m, 0.025], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.025000000000000001

   1. Initial program 35.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 78.9%

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

   if 0.025000000000000001 < k

   1. Initial program 28.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 79.0%

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

   9. Applied rewrites 78.4%

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

Alternative 6: 75.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\ell \cdot \cos k\_m}\\ \mathbf{if}\;k\_m \leq 8 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k\_m \cdot k\_m\right) \cdot t, -0.3333333333333333, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot k\_m}{\ell} \cdot t\_1}\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \frac{k\_m}{\ell}\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* l (cos k_m)))))
   (if (<= k_m 8e-141)
     (/
      2.0
      (*
       (/
        (* (* (* (fma (* (* k_m k_m) t) -0.3333333333333333 t) k_m) k_m) k_m)
        l)
       t_1))
     (if (<= k_m 2.5e+44)
       (/ 2.0 (* (* t (* (* k_m k_m) (/ k_m l))) t_1))
       (* (/ (/ (* l l) t) k_m) (/ -0.3333333333333333 k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (l * cos(k_m));
	double tmp;
	if (k_m <= 8e-141) {
		tmp = 2.0 / (((((fma(((k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * t_1);
	} else if (k_m <= 2.5e+44) {
		tmp = 2.0 / ((t * ((k_m * k_m) * (k_m / l))) * t_1);
	} else {
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(l * cos(k_m)))
	tmp = 0.0
	if (k_m <= 8e-141)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(k_m * k_m) * t), -0.3333333333333333, t) * k_m) * k_m) * k_m) / l) * t_1));
	elseif (k_m <= 2.5e+44)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k_m * k_m) * Float64(k_m / l))) * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / k_m) * Float64(-0.3333333333333333 / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8e-141], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * -0.3333333333333333 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.5e+44], N[(2.0 / N[(N[(t * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / k$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 8.0000000000000003e-141

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 76.7%

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

   if 8.0000000000000003e-141 < k < 2.4999999999999998e44

   1. Initial program 21.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 92.1

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 89.3%

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

   10. Applied rewrites 94.4%

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

   if 2.4999999999999998e44 < 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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 96.4

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{t} + \color{blue}{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t}}{{k}^{4}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}}{t}}{{k}^{4}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{t \cdot {k}^{4}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t \cdot {k}^{4}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Applied rewrites 16.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{t} \cdot \left(\ell \cdot \frac{\ell}{{k}^{4}}\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 55.8%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(\frac{k\_m}{\ell} \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.5e+44)
   (/ 2.0 (* (/ k_m (* (cos k_m) l)) (* (* k_m k_m) (* (/ k_m l) t))))
   (* (/ (/ (* l l) t) k_m) (/ -0.3333333333333333 k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e+44) {
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * ((k_m * k_m) * ((k_m / l) * t)));
	} else {
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d+44) then
        tmp = 2.0d0 / ((k_m / (cos(k_m) * l)) * ((k_m * k_m) * ((k_m / l) * t)))
    else
        tmp = (((l * l) / t) / k_m) * ((-0.3333333333333333d0) / k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e+44) {
		tmp = 2.0 / ((k_m / (Math.cos(k_m) * l)) * ((k_m * k_m) * ((k_m / l) * t)));
	} else {
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 2.5e+44:
		tmp = 2.0 / ((k_m / (math.cos(k_m) * l)) * ((k_m * k_m) * ((k_m / l) * t)))
	else:
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e+44)
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(Float64(k_m * k_m) * Float64(Float64(k_m / l) * t))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / k_m) * Float64(-0.3333333333333333 / k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e+44)
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * ((k_m * k_m) * ((k_m / l) * t)));
	else
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.5e+44], N[(2.0 / N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4999999999999998e44

   1. Initial program 34.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 80.6%

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

   if 2.4999999999999998e44 < 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

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 96.4

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{t} + \color{blue}{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t}}{{k}^{4}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}}{t}}{{k}^{4}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{t \cdot {k}^{4}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t \cdot {k}^{4}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Applied rewrites 16.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{t} \cdot \left(\ell \cdot \frac{\ell}{{k}^{4}}\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 55.8%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell}{k\_m \cdot k\_m}\\ \mathbf{if}\;k\_m \leq 8.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(k\_m \cdot k\_m, -0.3333333333333333, 2\right)}{t} \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ l (* k_m k_m))))
   (if (<= k_m 8.2e+106)
     (* (/ (fma (* k_m k_m) -0.3333333333333333 2.0) t) (* t_1 t_1))
     (* (/ (/ (* l l) t) k_m) (/ -0.3333333333333333 k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l / (k_m * k_m);
	double tmp;
	if (k_m <= 8.2e+106) {
		tmp = (fma((k_m * k_m), -0.3333333333333333, 2.0) / t) * (t_1 * t_1);
	} else {
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l / Float64(k_m * k_m))
	tmp = 0.0
	if (k_m <= 8.2e+106)
		tmp = Float64(Float64(fma(Float64(k_m * k_m), -0.3333333333333333, 2.0) / t) * Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / k_m) * Float64(-0.3333333333333333 / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 8.2e+106], N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / t), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / k$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.2000000000000005e106

   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 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 90.7

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

   5. Applied rewrites 90.7%

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

   7. Applied rewrites 95.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{t} + \color{blue}{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t}}{{k}^{4}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}}{t}}{{k}^{4}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{t \cdot {k}^{4}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t \cdot {k}^{4}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Applied rewrites 60.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{t} \cdot \left(\ell \cdot \frac{\ell}{{k}^{4}}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 67.3%

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

   if 8.2000000000000005e106 < k

   1. Initial program 33.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{t} + \color{blue}{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t}}{{k}^{4}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}}{t}}{{k}^{4}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{t \cdot {k}^{4}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t \cdot {k}^{4}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Applied rewrites 6.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{t} \cdot \left(\ell \cdot \frac{\ell}{{k}^{4}}\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.5%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.3% accurate, 9.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.55:\\ \;\;\;\;\frac{\ell}{\left(t \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{k\_m} \cdot \frac{-0.3333333333333333}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.55)
   (* (/ l (* (* t (* k_m k_m)) (* k_m k_m))) (* 2.0 l))
   (* (/ (/ (* l l) t) k_m) (/ -0.3333333333333333 k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.55) {
		tmp = (l / ((t * (k_m * k_m)) * (k_m * k_m))) * (2.0 * l);
	} else {
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / 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 <= 1.55d0) then
        tmp = (l / ((t * (k_m * k_m)) * (k_m * k_m))) * (2.0d0 * l)
    else
        tmp = (((l * l) / t) / k_m) * ((-0.3333333333333333d0) / 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 <= 1.55) {
		tmp = (l / ((t * (k_m * k_m)) * (k_m * k_m))) * (2.0 * l);
	} else {
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.55:
		tmp = (l / ((t * (k_m * k_m)) * (k_m * k_m))) * (2.0 * l)
	else:
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.55)
		tmp = Float64(Float64(l / Float64(Float64(t * Float64(k_m * k_m)) * Float64(k_m * k_m))) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / k_m) * Float64(-0.3333333333333333 / 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 <= 1.55)
		tmp = (l / ((t * (k_m * k_m)) * (k_m * k_m))) * (2.0 * l);
	else
		tmp = (((l * l) / t) / k_m) * (-0.3333333333333333 / 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, 1.55], N[(N[(l / N[(N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(-0.3333333333333333 / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004

   1. Initial program 35.4%

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

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 75.7%

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

   if 1.55000000000000004 < k

   1. Initial program 28.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{t}} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{k}^{4}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2}}{t} + \color{blue}{\frac{\frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      4. div-add-rev N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot {\ell}^{2} + \frac{-1}{3} \cdot \left({k}^{2} \cdot {\ell}^{2}\right)}{t}}}{{k}^{4}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\frac{2 \cdot {\ell}^{2} + \color{blue}{\left(\frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t}}{{k}^{4}} \]

      6. distribute-rgt-in N/A

         \[\leadsto \frac{\frac{\color{blue}{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}}{t}}{{k}^{4}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left(2 + \frac{-1}{3} \cdot {k}^{2}\right)}{t \cdot {k}^{4}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(2 + \frac{-1}{3} \cdot {k}^{2}\right) \cdot {\ell}^{2}}}{t \cdot {k}^{4}} \]

      9. times-frac N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{3} \cdot {k}^{2}}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]

   10. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{t} \cdot \left(\ell \cdot \frac{\ell}{{k}^{4}}\right)} \]

   11. Taylor expanded in k around inf

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 58.7%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{-0.3333333333333333}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.2% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 65.9

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

5. Applied rewrites 65.9%

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

7. Applied rewrites 68.5%

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

Alternative 11: 20.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) * (-0.11666666666666667 / 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 = (l * l) * ((-0.11666666666666667d0) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * l) * (-0.11666666666666667 / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

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

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k\right) \cdot k, k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 22.5%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
9. Step-by-step derivation

10. Applied rewrites 22.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{\color{blue}{t}} \]
11. Add Preprocessing

Alternative 12: 17.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \left(\ell \cdot \frac{-0.11666666666666667}{t}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m) :precision binary64 (* l (* l (/ -0.11666666666666667 t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l * (-0.11666666666666667 / 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 = l * (l * ((-0.11666666666666667d0) / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l * (-0.11666666666666667 / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l * (-0.11666666666666667 / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l * Float64(-0.11666666666666667 / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l * (-0.11666666666666667 / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right) \cdot k\right) \cdot k, k \cdot k, \frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)\right)}{{k}^{4}}} \]

6. Taylor expanded in k around inf

   \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

8. Applied rewrites 22.5%

   \[\leadsto \frac{\ell \cdot \ell}{t} \cdot \color{blue}{-0.11666666666666667} \]
9. Step-by-step derivation

10. Applied rewrites 20.3%

    \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot \color{blue}{-0.11666666666666667}\right) \]
11. Step-by-step derivation

12. Applied rewrites 20.3%

    \[\leadsto \ell \cdot \left(\ell \cdot \frac{-0.11666666666666667}{\color{blue}{t}}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))))