Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 98.8%

Time: 15.4s

Alternatives: 14

Speedup: 9.6×

• 38.2% 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: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 89.3%

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

7. Applied rewrites 95.1%

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 87.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \sin k\right) \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 8e-18)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (* (/ k l) t) (/ k l)) (sin k)) (tan k))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 8e-18) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k / l) * t) * (k / l)) * sin(k)) * tan(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 <= 8d-18) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((k / l) * t) * (k / l)) * sin(k)) * tan(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 <= 8e-18) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k / l) * t) * (k / l)) * Math.sin(k)) * Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 8e-18:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((((k / l) * t) * (k / l)) * math.sin(k)) * math.tan(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 8e-18)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k / l) * t) * Float64(k / l)) * sin(k)) * tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 8e-18)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((k / l) * t) * (k / l)) * sin(k)) * tan(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, 8e-18], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if k < 8.0000000000000006e-18

   1. Initial program 31.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 66.4%

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

   9. Applied rewrites 77.1%

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

   if 8.0000000000000006e-18 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 91.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.5%

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

   11. Applied rewrites 99.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k}{\ell} \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \sin k\right) \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 2.7e-13)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (* (/ (sin k) l) k) (/ k l)) t) (tan k))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 2.7e-13) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((sin(k) / l) * k) * (k / l)) * t) * tan(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 <= 2.7d-13) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((sin(k) / l) * k) * (k / l)) * t) * tan(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 <= 2.7e-13) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((Math.sin(k) / l) * k) * (k / l)) * t) * Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 2.7e-13:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((((math.sin(k) / l) * k) * (k / l)) * t) * math.tan(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 2.7e-13)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) / l) * k) * Float64(k / l)) * t) * tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 2.7e-13)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((sin(k) / l) * k) * (k / l)) * t) * tan(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, 2.7e-13], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000011e-13

   1. Initial program 31.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 77.2%

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

   if 2.70000000000000011e-13 < k

   1. Initial program 23.6%

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

   3. Taylor expanded in 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 91.5%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.5%

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

   11. Applied rewrites 97.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 2.7e-13)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (/ (* (* (* k t) (* (tan k) (sin k))) k) (* l l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 2.7e-13) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * t) * (tan(k) * sin(k))) * k) / (l * l));
	}
	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 <= 2.7d-13) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((k * t) * (tan(k) * sin(k))) * k) / (l * l))
    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 <= 2.7e-13) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * t) * (Math.tan(k) * Math.sin(k))) * k) / (l * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 2.7e-13:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((k * t) * (math.tan(k) * math.sin(k))) * k) / (l * l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 2.7e-13)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) * Float64(tan(k) * sin(k))) * k) / Float64(l * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 2.7e-13)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((k * t) * (tan(k) * sin(k))) * k) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 2.7e-13], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000011e-13

   1. Initial program 31.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 77.2%

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

   if 2.70000000000000011e-13 < k

   1. Initial program 23.6%

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

   3. Taylor expanded in 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 91.5%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 72.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot k\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 2.7e-13)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (/ (* (* (* (tan k) (sin k)) k) (* k t)) (* l l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 2.7e-13) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((tan(k) * sin(k)) * k) * (k * t)) / (l * l));
	}
	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 <= 2.7d-13) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((tan(k) * sin(k)) * k) * (k * t)) / (l * l))
    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 <= 2.7e-13) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((Math.tan(k) * Math.sin(k)) * k) * (k * t)) / (l * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 2.7e-13:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((math.tan(k) * math.sin(k)) * k) * (k * t)) / (l * l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 2.7e-13)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * k) * Float64(k * t)) / Float64(l * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 2.7e-13)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((tan(k) * sin(k)) * k) * (k * t)) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 2.7e-13], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000011e-13

   1. Initial program 31.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 66.5%

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

   9. Applied rewrites 77.2%

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

   if 2.70000000000000011e-13 < k

   1. Initial program 23.6%

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

   3. Taylor expanded in 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 91.5%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 72.3%

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

   11. Applied rewrites 72.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\tan k \cdot \sin k\right) \cdot k\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 89.3%

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

7. Applied rewrites 95.1%

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 7: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 89.3%

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

7. Applied rewrites 95.1%

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 98.1%

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

12. Final simplification 98.1%

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

Alternative 8: 75.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= (* l l) 5e+94)
     (/ 2.0 (* (* t_1 t) t_1))
     (/
      2.0
      (*
       (*
        (*
         (fma
          (* (/ t l) (fma 0.044444444444444446 (* k k) -0.3333333333333333))
          (* k k)
          (/ t l))
         (* k k))
        k)
       (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e+94) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((fma(((t / l) * fma(0.044444444444444446, (k * k), -0.3333333333333333)), (k * k), (t / l)) * (k * k)) * k) * ((k / cos(k)) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+94)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(Float64(t / l) * fma(0.044444444444444446, Float64(k * k), -0.3333333333333333)), Float64(k * k), Float64(t / l)) * Float64(k * k)) * k) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e+94], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t / l), $MachinePrecision] * N[(0.044444444444444446 * N[(k * k), $MachinePrecision] + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000001e94

   1. Initial program 30.7%

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

   3. Taylor expanded in 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 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 88.7%

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

   if 5.0000000000000001e94 < (*.f64 l l)

   1. Initial program 26.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. 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 89.4%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left({k}^{3} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{t}{\ell} + \frac{2}{45} \cdot \frac{{k}^{2} \cdot t}{\ell}\right) + \frac{t}{\ell}\right)}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

      \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\mathsf{fma}\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, \frac{t}{\ell}\right) \cdot \color{blue}{{k}^{3}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.3%

       \[\leadsto \frac{2}{\frac{\frac{k}{\cos k}}{\ell} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right) \cdot \frac{t}{\ell}, k \cdot k, \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(0.044444444444444446, k \cdot k, -0.3333333333333333\right), k \cdot k, \frac{t}{\ell}\right) \cdot \left(k \cdot k\right)\right) \cdot k\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 34:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 34.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (* k k) t) (/ k l)) (/ (/ k (cos k)) l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 34.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * ((k / cos(k)) / l));
	}
	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 <= 34.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((k * k) * t) * (k / l)) * ((k / cos(k)) / l))
    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 <= 34.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * ((k / Math.cos(k)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 34.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * ((k / math.cos(k)) / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 34.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t) * Float64(k / l)) * Float64(Float64(k / cos(k)) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 34.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((k * k) * t) * (k / l)) * ((k / cos(k)) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 34.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 34

   1. Initial program 30.7%

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

   3. Taylor expanded in 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 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 77.4%

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

   if 34 < k

   1. Initial program 25.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 58.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 34:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\ell}\right) \cdot \frac{\frac{k}{\cos k}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 34:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\cos k}}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 34.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (/ (* (* (* k k) t) (/ k (cos k))) l) (/ k l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 34.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k * k) * t) * (k / cos(k))) / l) * (k / l));
	}
	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 <= 34.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((k * k) * t) * (k / cos(k))) / l) * (k / l))
    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 <= 34.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k * k) * t) * (k / Math.cos(k))) / l) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 34.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((((k * k) * t) * (k / math.cos(k))) / l) * (k / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 34.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) * Float64(k / cos(k))) / l) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 34.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((k * k) * t) * (k / cos(k))) / l) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 34.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 34

   1. Initial program 30.7%

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

   3. Taylor expanded in 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 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 67.0%

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

   9. Applied rewrites 77.4%

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

   if 34 < k

   1. Initial program 25.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 58.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 34:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{k}{\cos k}}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.8% 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 29.0%

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

3. Taylor expanded in 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 62.3

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

5. Applied rewrites 62.3%

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

7. Applied rewrites 63.6%

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

9. Applied rewrites 70.9%

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

10. Final simplification 70.9%

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

Alternative 12: 74.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in 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 62.3

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

5. Applied rewrites 62.3%

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

7. Applied rewrites 63.6%

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

9. Applied rewrites 65.8%

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

11. Applied rewrites 66.5%

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

12. Final simplification 66.5%

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

Alternative 13: 69.7% accurate, 8.6× speedup

Math

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

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(k / l) / l) * Float64(k * k)) * Float64(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[(k / l), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in 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 62.3

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

5. Applied rewrites 62.3%

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

7. Applied rewrites 63.6%

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

9. Applied rewrites 65.8%

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

11. Applied rewrites 62.7%

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

12. Final simplification 62.7%

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

Alternative 14: 65.6% accurate, 9.6× speedup

Math

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

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(k / Float64(l * l)) * k) * k) * Float64(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[(k / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.0%

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

3. Taylor expanded in 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 62.3

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

5. Applied rewrites 62.3%

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

7. Applied rewrites 63.6%

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

9. Applied rewrites 65.8%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 58.6%

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

13. Final simplification 58.6%

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

Reproduce

herbie shell --seed 2024249 
(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))))