Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 98.6%

Time: 13.6s

Alternatives: 9

Speedup: 9.6×

• 39.5% 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: 35.6% 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: 98.6% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ (sin k) l) k) (* (* (/ k l) t) (tan k)))))

C

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

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 / (((sin(k) / l) * k) * (((k / l) * t) * tan(k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 89.5%

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

7. Applied rewrites 97.5%

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 85.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{k}{\ell} \cdot t}{\ell} \cdot \tan k\right) \cdot \sin k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 3.8e-106)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (/ (* (/ k l) t) l) (tan k)) (sin k)) k)))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 3.8e-106) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((((k / l) * t) / l) * tan(k)) * sin(k)) * k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 3.8d-106) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((((k / l) * t) / l) * tan(k)) * sin(k)) * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 3.8e-106) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((((k / l) * t) / l) * Math.tan(k)) * Math.sin(k)) * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 3.8e-106:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((((k / l) * t) / l) * math.tan(k)) * math.sin(k)) * k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 3.8e-106)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k / l) * t) / l) * tan(k)) * sin(k)) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 3.8e-106)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((((k / l) * t) / l) * tan(k)) * sin(k)) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 3.8e-106], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7999999999999999e-106

   1. Initial program 32.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.0

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 73.3%

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

   11. Applied rewrites 79.2%

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

   if 3.7999999999999999e-106 < k

   1. Initial program 28.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 96.3%

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

   11. Applied rewrites 96.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{k}{\ell} \cdot t}{\ell} \cdot \tan k\right) \cdot \sin k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot t}{\ell \cdot \ell}\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 6e-6)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (sin k) (tan k)) (/ (* k t) (* l l))) k)))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 6e-6) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((sin(k) * tan(k)) * ((k * t) / (l * l))) * k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 6d-6) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((sin(k) * tan(k)) * ((k * t) / (l * l))) * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 6e-6) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * ((k * t) / (l * l))) * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 6e-6:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((math.sin(k) * math.tan(k)) * ((k * t) / (l * l))) * k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 6e-6)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k * t) / Float64(l * l))) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 6e-6)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((sin(k) * tan(k)) * ((k * t) / (l * l))) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 6e-6], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.0000000000000002e-6

   1. Initial program 31.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 61.0%

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

   9. Applied rewrites 73.4%

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

   11. Applied rewrites 81.4%

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

   if 6.0000000000000002e-6 < k

   1. Initial program 28.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 95.0%

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

   11. Applied rewrites 86.6%

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

4. Final simplification 82.6%

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

Alternative 4: 97.2% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (* (* (/ k l) t) (/ k l)) (sin k)) (tan k))))

C

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

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 / (((((k / l) * t) * (k / l)) * sin(k)) * tan(k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 89.5%

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

7. Applied rewrites 97.5%

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

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

Alternative 5: 76.0% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k))) (/ 2.0 (* (* t_1 t) t_1))))

C

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

Fortran

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

Java

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

Python

def code(t, l, k):
	t_1 = (k / l) * k
	return 2.0 / ((t_1 * t) * t_1)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 70.0

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

5. Applied rewrites 70.0%

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

7. Applied rewrites 61.6%

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

9. Applied rewrites 71.6%

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

11. Applied rewrites 77.8%

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

12. Final simplification 77.8%

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

Alternative 6: 71.5% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (* (/ k l) (/ k l)) (* k k)) t)))

C

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

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 / ((((k / l) * (k / l)) * (k * k)) * t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((k / l) * (k / l)) * (k * k)) * t);
}

Python

def code(t, l, k):
	return 2.0 / ((((k / l) * (k / l)) * (k * k)) * t)

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 70.0

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

5. Applied rewrites 70.0%

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

7. Applied rewrites 61.6%

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

9. Applied rewrites 71.6%

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

10. Final simplification 71.6%

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

Alternative 7: 69.8% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (* (/ (/ k l) l) k) (* k k)) t)))

C

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

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 / (((((k / l) / l) * k) * (k * k)) * t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((((k / l) / l) * k) * (k * k)) * t);
}

Python

def code(t, l, k):
	return 2.0 / (((((k / l) / l) * k) * (k * k)) * t)

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[(k / l), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 70.0

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

5. Applied rewrites 70.0%

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

7. Applied rewrites 61.6%

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

9. Applied rewrites 71.6%

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

11. Applied rewrites 67.7%

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

12. Final simplification 67.7%

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

Alternative 8: 64.4% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ (* k k) (* l l)) (* k k)) t)))

C

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

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 / ((((k * k) / (l * l)) * (k * k)) * t)
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((k * k) / (l * l)) * (k * k)) * t);
}

Python

def code(t, l, k):
	return 2.0 / ((((k * k) / (l * l)) * (k * k)) * t)

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 70.0

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

5. Applied rewrites 70.0%

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

7. Applied rewrites 61.6%

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

8. Final simplification 61.6%

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

Alternative 9: 60.9% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* l l) t) (/ 2.0 (* (* k k) (* k k)))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return ((l * l) / t) * (2.0 / ((k * k) * (k * k)));
}

Python

def code(t, l, k):
	return ((l * l) / t) * (2.0 / ((k * k) * (k * k)))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. cube-mult N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 65.1%

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

5. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 59.8

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

7. Applied rewrites 59.8%

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

9. Applied rewrites 59.8%

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

10. Final simplification 59.8%

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

Reproduce

herbie shell --seed 2024251 
(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))))