Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 97.0%

Time: 12.4s

Alternatives: 15

Speedup: 11.0×

• 40.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 35.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{k\_m}{\cos k\_m \cdot \ell}\\ \mathbf{if}\;k\_m \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot t, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot t\right)\right) \cdot \frac{k\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ k_m (* (cos k_m) l))))
   (if (<= k_m 5.8e-11)
     (/
      2.0
      (*
       t_1
       (*
        (* (* (fma -0.3333333333333333 (* (* k_m k_m) t) t) k_m) k_m)
        (/ k_m l))))
     (/ 2.0 (* (* t_1 (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) t)) (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m / (cos(k_m) * l);
	double tmp;
	if (k_m <= 5.8e-11) {
		tmp = 2.0 / (t_1 * (((fma(-0.3333333333333333, ((k_m * k_m) * t), t) * k_m) * k_m) * (k_m / l)));
	} else {
		tmp = 2.0 / ((t_1 * ((0.5 - (0.5 * cos((k_m + k_m)))) * t)) * (k_m / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m / Float64(cos(k_m) * l))
	tmp = 0.0
	if (k_m <= 5.8e-11)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(k_m * k_m) * t), t) * k_m) * k_m) * Float64(k_m / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(k_m + k_m)))) * t)) * Float64(k_m / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 5.8e-11], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(-0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e-11

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0

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

      1. *-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 95.0

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

   5. Applied rewrites 95.0%

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

   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 79.7%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.8%

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

   if 5.8e-11 < k

   1. Initial program 32.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.3%

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

Alternative 2: 96.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 94.3

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

5. Applied rewrites 94.3%

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

7. Applied rewrites 98.3%

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

9. Applied rewrites 98.8%

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

Alternative 3: 96.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 94.3

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

5. Applied rewrites 94.3%

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

7. Applied rewrites 98.8%

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

Alternative 4: 96.4% accurate, 1.3× speedup

Math

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

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) t)) (/ k_m l))))

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) * t)) * (k_m / l));
}

Fortran

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

Java

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

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) * t)) * (k_m / l))

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)) * Float64((sin(k_m) ^ 2.0) * t)) * Float64(k_m / l)))
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) * t)) * (k_m / l));
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[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 94.3

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

5. Applied rewrites 94.3%

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

7. Applied rewrites 98.3%

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

Alternative 5: 97.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{\cos k\_m \cdot \ell} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, \left(k\_m \cdot k\_m\right) \cdot t, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - 0.5 \cdot \cos \left(k\_m + k\_m\right)\right) \cdot \left(t \cdot \frac{k\_m}{\ell}\right)\right) \cdot \frac{k\_m}{\ell \cdot \cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.8e-11)
   (/
    2.0
    (*
     (/ k_m (* (cos k_m) l))
     (*
      (* (* (fma -0.3333333333333333 (* (* k_m k_m) t) t) k_m) k_m)
      (/ k_m l))))
   (/
    2.0
    (*
     (* (- 0.5 (* 0.5 (cos (+ k_m k_m)))) (* t (/ k_m l)))
     (/ k_m (* l (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.8e-11) {
		tmp = 2.0 / ((k_m / (cos(k_m) * l)) * (((fma(-0.3333333333333333, ((k_m * k_m) * t), t) * k_m) * k_m) * (k_m / l)));
	} else {
		tmp = 2.0 / (((0.5 - (0.5 * cos((k_m + k_m)))) * (t * (k_m / l))) * (k_m / (l * cos(k_m))));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.8e-11)
		tmp = Float64(2.0 / Float64(Float64(k_m / Float64(cos(k_m) * l)) * Float64(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(k_m * k_m) * t), t) * k_m) * k_m) * Float64(k_m / l))));
	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))) * Float64(k_m / Float64(l * cos(k_m)))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.8e-11], N[(2.0 / N[(N[(k$95$m / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $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] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e-11

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0

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

      1. *-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 95.0

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

   5. Applied rewrites 95.0%

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

   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 79.7%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.8%

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

   if 5.8e-11 < k

   1. Initial program 32.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.2%

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

Alternative 6: 87.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 94.3

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

5. Applied rewrites 94.3%

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

7. Applied rewrites 90.9%

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

Alternative 7: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \left(k\_m \cdot k\_m\right) \cdot t\\ t_2 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{t\_2} \cdot \left(\left(\left(\mathsf{fma}\left(-0.3333333333333333, t\_1, t\right) \cdot k\_m\right) \cdot k\_m\right) \cdot \frac{k\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1} \cdot \frac{t\_2 \cdot \ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (* k_m k_m) t)) (t_2 (* (cos k_m) l)))
   (if (<= k_m 5.8e-11)
     (/
      2.0
      (*
       (/ k_m t_2)
       (* (* (* (fma -0.3333333333333333 t_1 t) k_m) k_m) (/ k_m l))))
     (* (/ 2.0 t_1) (/ (* t_2 l) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double t_2 = cos(k_m) * l;
	double tmp;
	if (k_m <= 5.8e-11) {
		tmp = 2.0 / ((k_m / t_2) * (((fma(-0.3333333333333333, t_1, t) * k_m) * k_m) * (k_m / l)));
	} else {
		tmp = (2.0 / t_1) * ((t_2 * l) / (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m * k_m) * t)
	t_2 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 5.8e-11)
		tmp = Float64(2.0 / Float64(Float64(k_m / t_2) * Float64(Float64(Float64(fma(-0.3333333333333333, t_1, t) * k_m) * k_m) * Float64(k_m / l))));
	else
		tmp = Float64(Float64(2.0 / t_1) * Float64(Float64(t_2 * l) / Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 5.8e-11], N[(2.0 / N[(N[(k$95$m / t$95$2), $MachinePrecision] * N[(N[(N[(N[(-0.3333333333333333 * t$95$1 + t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$1), $MachinePrecision] * N[(N[(t$95$2 * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8e-11

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0

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

      1. *-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 95.0

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

   5. Applied rewrites 95.0%

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

   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 79.7%

      \[\leadsto \frac{2}{\frac{\left(\left(\mathsf{fma}\left(\left(k \cdot k\right) \cdot t, -0.3333333333333333, t\right) \cdot k\right) \cdot k\right) \cdot k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.8%

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

   if 5.8e-11 < k

   1. Initial program 32.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   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 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 74.9

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

   8. Applied rewrites 74.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 56.5%

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

Alternative 8: 74.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3450000000000:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\_m\right) \cdot k\_m}}{k\_m}}{k\_m} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot \frac{\left(\cos k\_m \cdot \ell\right) \cdot \ell}{k\_m \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3450000000000.0)
   (* (/ (/ (/ l (* (* t k_m) k_m)) k_m) k_m) (* 2.0 l))
   (* (/ 2.0 (* (* k_m k_m) t)) (/ (* (* (cos k_m) l) l) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3450000000000.0) {
		tmp = (((l / ((t * k_m) * k_m)) / k_m) / k_m) * (2.0 * l);
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * (((cos(k_m) * l) * l) / (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3450000000000.0d0) then
        tmp = (((l / ((t * k_m) * k_m)) / k_m) / k_m) * (2.0d0 * l)
    else
        tmp = (2.0d0 / ((k_m * k_m) * t)) * (((cos(k_m) * l) * l) / (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3450000000000.0) {
		tmp = (((l / ((t * k_m) * k_m)) / k_m) / k_m) * (2.0 * l);
	} else {
		tmp = (2.0 / ((k_m * k_m) * t)) * (((Math.cos(k_m) * l) * l) / (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3450000000000.0:
		tmp = (((l / ((t * k_m) * k_m)) / k_m) / k_m) * (2.0 * l)
	else:
		tmp = (2.0 / ((k_m * k_m) * t)) * (((math.cos(k_m) * l) * l) / (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3450000000000.0)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k_m) * k_m)) / k_m) / k_m) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k_m * k_m) * t)) * Float64(Float64(Float64(cos(k_m) * l) * l) / Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3450000000000.0)
		tmp = (((l / ((t * k_m) * k_m)) / k_m) / k_m) * (2.0 * l);
	else
		tmp = (2.0 / ((k_m * k_m) * t)) * (((cos(k_m) * l) * l) / (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3450000000000.0], N[(N[(N[(N[(l / N[(N[(t * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.45e12

   1. Initial program 42.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 79.0%

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

   9. Applied rewrites 79.0%

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

   11. Applied rewrites 81.8%

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

   if 3.45e12 < k

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 75.2

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

   8. Applied rewrites 75.2%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 57.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3450000000000:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.0% accurate, 2.9× speedup

Math

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

FPCore

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

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (((k_m * k_m) * (t * (k_m / l))) * (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(k_m * k_m) * Float64(t * Float64(k_m / l))) * 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 * k_m) * (t * (k_m / l))) * (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[(k$95$m * k$95$m), $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 94.3

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

5. Applied rewrites 94.3%

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 76.4%

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

Alternative 10: 73.1% accurate, 8.6× speedup

Math

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

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(Float64(Float64(l / Float64(Float64(t * k_m) * k_m)) / 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[(N[(N[(l / N[(N[(t * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.6

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

5. Applied rewrites 72.6%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

11. Applied rewrites 75.4%

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

12. Final simplification 75.4%

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

Alternative 11: 72.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.6

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

5. Applied rewrites 72.6%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 75.3%

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

Alternative 12: 72.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.6

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

5. Applied rewrites 72.6%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 74.9%

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

Alternative 13: 73.2% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.6

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

5. Applied rewrites 72.6%

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

7. Applied rewrites 74.9%

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

Alternative 14: 70.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 72.6

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

5. Applied rewrites 72.6%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

11. Applied rewrites 73.4%

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

Alternative 15: 28.5% accurate, 12.5× speedup

Math

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

FPCore

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

C

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

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (2.0d0 / ((k_m * k_m) * t)) * ((-0.16666666666666666d0) * (l * l))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (2.0 / ((k_m * k_m) * t)) * (-0.16666666666666666 * (l * l));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (2.0 / ((k_m * k_m) * t)) * (-0.16666666666666666 * (l * l))

Julia

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

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (2.0 / ((k_m * k_m) * t)) * (-0.16666666666666666 * (l * l));
end

Wolfram

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

Derivation

1. Initial program 40.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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 94.3

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

5. Applied rewrites 94.3%

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

6. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lower-sin.f64 77.3

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

8. Applied rewrites 77.3%

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

9. Taylor expanded in k around 0

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

11. Applied rewrites 44.1%

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

12. Taylor expanded in k around inf

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

14. Applied rewrites 28.9%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot \color{blue}{\ell}\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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))))