Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 98.5%

Time: 13.9s

Alternatives: 15

Speedup: 9.6×

• 39.9% 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: 35.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in t around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 92.2%

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

7. Applied rewrites 94.7%

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

9. Applied rewrites 98.8%

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

Alternative 2: 86.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \cos k \cdot \ell\\ \mathbf{if}\;k \leq 2.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\frac{k}{\ell} \cdot \left(t\_1 \cdot t\right)}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t\_1 \cdot k}{\ell} \cdot \frac{k}{t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)) (t_2 (* (cos k) l)))
   (if (<= k 2.7e-34)
     (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
     (if (<= k 6.6e+140)
       (/ 2.0 (* k (/ (* (/ k l) (* t_1 t)) t_2)))
       (/ 2.0 (* t (* (/ (* t_1 k) l) (/ k t_2))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = cos(k) * l;
	double tmp;
	if (k <= 2.7e-34) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 6.6e+140) {
		tmp = 2.0 / (k * (((k / l) * (t_1 * t)) / t_2));
	} else {
		tmp = 2.0 / (t * (((t_1 * k) / l) * (k / t_2)));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = cos(k) * l
    if (k <= 2.7d-34) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else if (k <= 6.6d+140) then
        tmp = 2.0d0 / (k * (((k / l) * (t_1 * t)) / t_2))
    else
        tmp = 2.0d0 / (t * (((t_1 * k) / l) * (k / t_2)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = Math.cos(k) * l;
	double tmp;
	if (k <= 2.7e-34) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 6.6e+140) {
		tmp = 2.0 / (k * (((k / l) * (t_1 * t)) / t_2));
	} else {
		tmp = 2.0 / (t * (((t_1 * k) / l) * (k / t_2)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	t_2 = math.cos(k) * l
	tmp = 0
	if k <= 2.7e-34:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	elif k <= 6.6e+140:
		tmp = 2.0 / (k * (((k / l) * (t_1 * t)) / t_2))
	else:
		tmp = 2.0 / (t * (((t_1 * k) / l) * (k / t_2)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(cos(k) * l)
	tmp = 0.0
	if (k <= 2.7e-34)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	elseif (k <= 6.6e+140)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k / l) * Float64(t_1 * t)) / t_2)));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(t_1 * k) / l) * Float64(k / t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	t_2 = cos(k) * l;
	tmp = 0.0;
	if (k <= 2.7e-34)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	elseif (k <= 6.6e+140)
		tmp = 2.0 / (k * (((k / l) * (t_1 * t)) / t_2));
	else
		tmp = 2.0 / (t * (((t_1 * k) / l) * (k / t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k, 2.7e-34], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.6e+140], N[(2.0 / N[(k * N[(N[(N[(k / l), $MachinePrecision] * N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(t$95$1 * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.70000000000000017e-34

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 86.1%

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

   if 2.70000000000000017e-34 < k < 6.6000000000000003e140

   1. Initial program 15.4%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 96.1%

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

   if 6.6000000000000003e140 < k

   1. Initial program 43.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 96.5

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

   7. Applied rewrites 96.5%

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

Alternative 3: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell \cdot \cos k} \cdot \frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-34)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (/ 2.0 (* (/ k (* l (cos k))) (/ (* (* (pow (sin k) 2.0) t) k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-34) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / ((k / (l * cos(k))) * (((pow(sin(k), 2.0) * t) * 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) :: tmp
    if (k <= 2.7d-34) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else
        tmp = 2.0d0 / ((k / (l * cos(k))) * ((((sin(k) ** 2.0d0) * t) * k) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-34) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / ((k / (l * Math.cos(k))) * (((Math.pow(Math.sin(k), 2.0) * t) * k) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-34:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	else:
		tmp = 2.0 / ((k / (l * math.cos(k))) * (((math.pow(math.sin(k), 2.0) * t) * k) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-34)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l * cos(k))) * Float64(Float64(Float64((sin(k) ^ 2.0) * t) * k) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-34)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	else
		tmp = 2.0 / ((k / (l * cos(k))) * ((((sin(k) ^ 2.0) * t) * k) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-34], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000017e-34

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 86.1%

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

   if 2.70000000000000017e-34 < k

   1. Initial program 27.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 92.6%

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

   7. Applied rewrites 92.5%

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

Alternative 4: 86.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-34)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (/ 2.0 (* k (/ (* (/ k l) (* (pow (sin k) 2.0) t)) (* (cos k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-34) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / (k * (((k / l) * (pow(sin(k), 2.0) * t)) / (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) :: tmp
    if (k <= 2.7d-34) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else
        tmp = 2.0d0 / (k * (((k / l) * ((sin(k) ** 2.0d0) * t)) / (cos(k) * l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-34) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / (k * (((k / l) * (Math.pow(Math.sin(k), 2.0) * t)) / (Math.cos(k) * l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-34:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	else:
		tmp = 2.0 / (k * (((k / l) * (math.pow(math.sin(k), 2.0) * t)) / (math.cos(k) * l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-34)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k / l) * Float64((sin(k) ^ 2.0) * t)) / Float64(cos(k) * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-34)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	else
		tmp = 2.0 / (k * (((k / l) * ((sin(k) ^ 2.0) * t)) / (cos(k) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-34], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(k / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000017e-34

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 65.3%

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

   9. Applied rewrites 86.1%

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

   if 2.70000000000000017e-34 < k

   1. Initial program 27.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 92.6%

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

   7. Applied rewrites 92.5%

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

   9. Applied rewrites 96.3%

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

Alternative 5: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\ell \cdot {\sin k}^{-2}\right) \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\tan k \cdot \sin k\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.8e-14)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (if (<= k 1.6e+141)
     (* (/ (* 2.0 (cos k)) (* (* k k) t)) (* (* l (pow (sin k) -2.0)) l))
     (/
      2.0
      (*
       t
       (* (* (/ t l) (/ t l)) (* (/ k t) (* (/ k t) (* (tan k) (sin k))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.8e-14) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 1.6e+141) {
		tmp = ((2.0 * cos(k)) / ((k * k) * t)) * ((l * pow(sin(k), -2.0)) * l);
	} else {
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(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) :: tmp
    if (k <= 3.8d-14) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else if (k <= 1.6d+141) then
        tmp = ((2.0d0 * cos(k)) / ((k * k) * t)) * ((l * (sin(k) ** (-2.0d0))) * l)
    else
        tmp = 2.0d0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.8e-14) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 1.6e+141) {
		tmp = ((2.0 * Math.cos(k)) / ((k * k) * t)) * ((l * Math.pow(Math.sin(k), -2.0)) * l);
	} else {
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (Math.tan(k) * Math.sin(k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.8e-14:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	elif k <= 1.6e+141:
		tmp = ((2.0 * math.cos(k)) / ((k * k) * t)) * ((l * math.pow(math.sin(k), -2.0)) * l)
	else:
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (math.tan(k) * math.sin(k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.8e-14)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	elseif (k <= 1.6e+141)
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(k * k) * t)) * Float64(Float64(l * (sin(k) ^ -2.0)) * l));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(t / l) * Float64(t / l)) * Float64(Float64(k / t) * Float64(Float64(k / t) * Float64(tan(k) * sin(k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.8e-14)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	elseif (k <= 1.6e+141)
		tmp = ((2.0 * cos(k)) / ((k * k) * t)) * ((l * (sin(k) ^ -2.0)) * l);
	else
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.8e-14], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+141], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.8000000000000002e-14

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 86.5%

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

   if 3.8000000000000002e-14 < k < 1.60000000000000009e141

   1. Initial program 12.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 85.1

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

   8. Applied rewrites 85.1%

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

   10. Applied rewrites 85.1%

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

   if 1.60000000000000009e141 < k

   1. Initial program 43.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 77.2

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

   6. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   8. Applied rewrites 80.0%

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

Alternative 6: 82.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\left(k \cdot {\sin k}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-30)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (/ 2.0 (* k (/ (* (* k (pow (sin k) 2.0)) t) (* (* (cos k) l) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-30) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / (k * (((k * pow(sin(k), 2.0)) * t) / ((cos(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) :: tmp
    if (k <= 2.7d-30) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else
        tmp = 2.0d0 / (k * (((k * (sin(k) ** 2.0d0)) * t) / ((cos(k) * l) * l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-30) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / (k * (((k * Math.pow(Math.sin(k), 2.0)) * t) / ((Math.cos(k) * l) * l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-30:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	else:
		tmp = 2.0 / (k * (((k * math.pow(math.sin(k), 2.0)) * t) / ((math.cos(k) * l) * l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-30)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k * (sin(k) ^ 2.0)) * t) / Float64(Float64(cos(k) * l) * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-30)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	else
		tmp = 2.0 / (k * (((k * (sin(k) ^ 2.0)) * t) / ((cos(k) * l) * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-30], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(k * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999987e-30

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 86.2%

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

   if 2.69999999999999987e-30 < k

   1. Initial program 26.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.3%

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

   9. Applied rewrites 84.8%

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

Alternative 7: 82.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot t\right) \cdot k} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-30)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (* (/ 2.0 (* (* (pow (sin k) 2.0) t) k)) (/ (* (* (cos k) l) l) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-30) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = (2.0 / ((pow(sin(k), 2.0) * t) * k)) * (((cos(k) * l) * l) / k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.7d-30) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else
        tmp = (2.0d0 / (((sin(k) ** 2.0d0) * t) * k)) * (((cos(k) * l) * l) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-30) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = (2.0 / ((Math.pow(Math.sin(k), 2.0) * t) * k)) * (((Math.cos(k) * l) * l) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-30:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	else:
		tmp = (2.0 / ((math.pow(math.sin(k), 2.0) * t) * k)) * (((math.cos(k) * l) * l) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-30)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * t) * k)) * Float64(Float64(Float64(cos(k) * l) * l) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-30)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	else
		tmp = (2.0 / (((sin(k) ^ 2.0) * t) * k)) * (((cos(k) * l) * l) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-30], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999987e-30

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 86.2%

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

   if 2.69999999999999987e-30 < k

   1. Initial program 26.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in t around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. lower-/.f64 N/A

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

   12. Applied rewrites 83.7%

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

Alternative 8: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;\left(\cos k \cdot 2\right) \cdot \frac{\ell \cdot \ell}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\tan k \cdot \sin k\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.55e-30)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (if (<= k 1.6e+141)
     (* (* (cos k) 2.0) (/ (* l l) (* t (pow (* (sin k) k) 2.0))))
     (/
      2.0
      (*
       t
       (* (* (/ t l) (/ t l)) (* (/ k t) (* (/ k t) (* (tan k) (sin k))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.55e-30) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 1.6e+141) {
		tmp = (cos(k) * 2.0) * ((l * l) / (t * pow((sin(k) * k), 2.0)));
	} else {
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(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) :: tmp
    if (k <= 2.55d-30) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else if (k <= 1.6d+141) then
        tmp = (cos(k) * 2.0d0) * ((l * l) / (t * ((sin(k) * k) ** 2.0d0)))
    else
        tmp = 2.0d0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.55e-30) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 1.6e+141) {
		tmp = (Math.cos(k) * 2.0) * ((l * l) / (t * Math.pow((Math.sin(k) * k), 2.0)));
	} else {
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (Math.tan(k) * Math.sin(k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.55e-30:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	elif k <= 1.6e+141:
		tmp = (math.cos(k) * 2.0) * ((l * l) / (t * math.pow((math.sin(k) * k), 2.0)))
	else:
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (math.tan(k) * math.sin(k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.55e-30)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	elseif (k <= 1.6e+141)
		tmp = Float64(Float64(cos(k) * 2.0) * Float64(Float64(l * l) / Float64(t * (Float64(sin(k) * k) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(t / l) * Float64(t / l)) * Float64(Float64(k / t) * Float64(Float64(k / t) * Float64(tan(k) * sin(k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.55e-30)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	elseif (k <= 1.6e+141)
		tmp = (cos(k) * 2.0) * ((l * l) / (t * ((sin(k) * k) ^ 2.0)));
	else
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.55e-30], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+141], N[(N[(N[Cos[k], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.54999999999999986e-30

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 65.1%

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

   9. Applied rewrites 86.2%

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

   if 2.54999999999999986e-30 < k < 1.60000000000000009e141

   1. Initial program 13.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 93.8%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 86.6

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

   8. Applied rewrites 86.6%

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

   10. Applied rewrites 86.5%

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

   if 1.60000000000000009e141 < k

   1. Initial program 43.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 77.2

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

   6. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   8. Applied rewrites 80.0%

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

Alternative 9: 81.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.000135:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\tan k \cdot \sin k\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.000135)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (if (<= k 1.6e+141)
     (*
      (/ (* 2.0 (cos k)) (* (* k k) t))
      (/ (* l l) (- 0.5 (* 0.5 (cos (+ k k))))))
     (/
      2.0
      (*
       t
       (* (* (/ t l) (/ t l)) (* (/ k t) (* (/ k t) (* (tan k) (sin k))))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000135) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 1.6e+141) {
		tmp = ((2.0 * cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * cos((k + k)))));
	} else {
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(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) :: tmp
    if (k <= 0.000135d0) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else if (k <= 1.6d+141) then
        tmp = ((2.0d0 * cos(k)) / ((k * k) * t)) * ((l * l) / (0.5d0 - (0.5d0 * cos((k + k)))))
    else
        tmp = 2.0d0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000135) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 1.6e+141) {
		tmp = ((2.0 * Math.cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * Math.cos((k + k)))));
	} else {
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (Math.tan(k) * Math.sin(k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.000135:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	elif k <= 1.6e+141:
		tmp = ((2.0 * math.cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * math.cos((k + k)))))
	else:
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (math.tan(k) * math.sin(k))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.000135)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	elseif (k <= 1.6e+141)
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(k * k) * t)) * Float64(Float64(l * l) / Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(Float64(t / l) * Float64(t / l)) * Float64(Float64(k / t) * Float64(Float64(k / t) * Float64(tan(k) * sin(k)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.000135)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	elseif (k <= 1.6e+141)
		tmp = ((2.0 * cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * cos((k + k)))));
	else
		tmp = 2.0 / (t * (((t / l) * (t / l)) * ((k / t) * ((k / t) * (tan(k) * sin(k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.000135], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+141], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(N[(k / t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.35000000000000002e-4

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 86.5%

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

   if 1.35000000000000002e-4 < k < 1.60000000000000009e141

   1. Initial program 12.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 85.1

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

   8. Applied rewrites 85.1%

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

   10. Applied rewrites 84.2%

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

   if 1.60000000000000009e141 < k

   1. Initial program 43.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 77.2

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

   6. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 80.0

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

   8. Applied rewrites 80.0%

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

Alternative 10: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.000135:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{0.5 - 0.5 \cdot \cos \left(k + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.000135)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (if (<= k 6.6e+140)
     (*
      (/ (* 2.0 (cos k)) (* (* k k) t))
      (/ (* l l) (- 0.5 (* 0.5 (cos (+ k k))))))
     (/ 2.0 (* (/ k l) (/ (* (* (pow (sin k) 2.0) t) k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000135) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 6.6e+140) {
		tmp = ((2.0 * cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * cos((k + k)))));
	} else {
		tmp = 2.0 / ((k / l) * (((pow(sin(k), 2.0) * t) * 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) :: tmp
    if (k <= 0.000135d0) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else if (k <= 6.6d+140) then
        tmp = ((2.0d0 * cos(k)) / ((k * k) * t)) * ((l * l) / (0.5d0 - (0.5d0 * cos((k + k)))))
    else
        tmp = 2.0d0 / ((k / l) * ((((sin(k) ** 2.0d0) * t) * k) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.000135) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else if (k <= 6.6e+140) {
		tmp = ((2.0 * Math.cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * Math.cos((k + k)))));
	} else {
		tmp = 2.0 / ((k / l) * (((Math.pow(Math.sin(k), 2.0) * t) * k) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.000135:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	elif k <= 6.6e+140:
		tmp = ((2.0 * math.cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * math.cos((k + k)))))
	else:
		tmp = 2.0 / ((k / l) * (((math.pow(math.sin(k), 2.0) * t) * k) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.000135)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	elseif (k <= 6.6e+140)
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(k * k) * t)) * Float64(Float64(l * l) / Float64(0.5 - Float64(0.5 * cos(Float64(k + k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(Float64((sin(k) ^ 2.0) * t) * k) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.000135)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	elseif (k <= 6.6e+140)
		tmp = ((2.0 * cos(k)) / ((k * k) * t)) * ((l * l) / (0.5 - (0.5 * cos((k + k)))));
	else
		tmp = 2.0 / ((k / l) * ((((sin(k) ^ 2.0) * t) * k) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.000135], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.6e+140], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.35000000000000002e-4

   1. Initial program 34.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 86.5%

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

   if 1.35000000000000002e-4 < k < 6.6000000000000003e140

   1. Initial program 12.2%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 85.1

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

   8. Applied rewrites 85.1%

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

   10. Applied rewrites 84.2%

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

   if 6.6000000000000003e140 < k

   1. Initial program 43.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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 90.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 67.7%

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

Alternative 11: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 24:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \frac{\left({\sin k}^{2} \cdot t\right) \cdot k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 24.0)
   (/ 2.0 (/ (* (* (/ k l) k) t) (/ (/ l k) k)))
   (/ 2.0 (* (/ k l) (/ (* (* (pow (sin k) 2.0) t) k) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 24.0) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / ((k / l) * (((pow(sin(k), 2.0) * t) * 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) :: tmp
    if (k <= 24.0d0) then
        tmp = 2.0d0 / ((((k / l) * k) * t) / ((l / k) / k))
    else
        tmp = 2.0d0 / ((k / l) * ((((sin(k) ** 2.0d0) * t) * k) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 24.0) {
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	} else {
		tmp = 2.0 / ((k / l) * (((Math.pow(Math.sin(k), 2.0) * t) * k) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 24.0:
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k))
	else:
		tmp = 2.0 / ((k / l) * (((math.pow(math.sin(k), 2.0) * t) * k) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 24.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * k) * t) / Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(Float64((sin(k) ^ 2.0) * t) * k) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 24.0)
		tmp = 2.0 / ((((k / l) * k) * t) / ((l / k) / k));
	else
		tmp = 2.0 / ((k / l) * ((((sin(k) ^ 2.0) * t) * k) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 24.0], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 24

   1. Initial program 33.7%

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

   3. Taylor expanded in 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 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 65.7%

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

   9. Applied rewrites 86.4%

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

   if 24 < k

   1. Initial program 27.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 52.1%

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

Alternative 12: 76.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in 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 68.6

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

5. Applied rewrites 68.6%

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

7. Applied rewrites 60.8%

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

9. Applied rewrites 76.7%

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

Alternative 13: 75.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in 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 68.6

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

5. Applied rewrites 68.6%

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

7. Applied rewrites 72.3%

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

9. Applied rewrites 74.5%

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

11. Applied rewrites 76.7%

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

Alternative 14: 75.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in 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 68.6

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

5. Applied rewrites 68.6%

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

7. Applied rewrites 72.3%

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

9. Applied rewrites 74.5%

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

11. Applied rewrites 76.7%

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

Alternative 15: 64.3% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.3%

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

3. Taylor expanded in 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 68.6

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

5. Applied rewrites 68.6%

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

7. Applied rewrites 60.8%

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

9. Applied rewrites 62.0%

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

Reproduce

herbie shell --seed 2024315 
(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))))