Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 98.0%

Time: 14.2s

Alternatives: 20

Speedup: 11.0×

• 39.4% 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: 35.5% 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: 98.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.9

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

5. Applied rewrites 91.9%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 97.7%

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

11. Applied rewrites 98.5%

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

Alternative 2: 95.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\sin k \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \left(\frac{k}{\ell} \cdot t\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (/ 2.0 (* (sin k) (* (sin k) (* (/ k (* (cos k) l)) (* (/ k l) t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = 2.0 / (sin(k) * (sin(k) * ((k / (cos(k) * l)) * ((k / l) * t))));
	}
	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 ((l * l) <= 0.0d0) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = 2.0d0 / (sin(k) * (sin(k) * ((k / (cos(k) * l)) * ((k / l) * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.sin(k) * ((k / (Math.cos(k) * l)) * ((k / l) * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = 2.0 / (math.sin(k) * (math.sin(k) * ((k / (math.cos(k) * l)) * ((k / l) * t))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 17.2%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 83.4%

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

   9. Applied rewrites 83.4%

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

   11. Applied rewrites 94.0%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 93.7

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 99.6%

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

Alternative 3: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-309}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \left(\frac{k}{\cos k \cdot \ell} \cdot \left(\frac{k}{\ell} \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-309)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (/ 2.0 (* (pow (sin k) 2.0) (* (/ k (* (cos k) l)) (* (/ k l) t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-309) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = 2.0 / (pow(sin(k), 2.0) * ((k / (cos(k) * l)) * ((k / l) * t)));
	}
	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 ((l * l) <= 1d-309) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = 2.0d0 / ((sin(k) ** 2.0d0) * ((k / (cos(k) * l)) * ((k / l) * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-309) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = 2.0 / (Math.pow(Math.sin(k), 2.0) * ((k / (Math.cos(k) * l)) * ((k / l) * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-309:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = 2.0 / (math.pow(math.sin(k), 2.0) * ((k / (math.cos(k) * l)) * ((k / l) * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-309)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(Float64(k / Float64(cos(k) * l)) * Float64(Float64(k / l) * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-309)
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	else
		tmp = 2.0 / ((sin(k) ^ 2.0) * ((k / (cos(k) * l)) * ((k / l) * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-309], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.000000000000002e-309

   1. Initial program 16.4%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 84.1%

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

   9. Applied rewrites 84.1%

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

   11. Applied rewrites 94.3%

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

   if 1.000000000000002e-309 < (*.f64 l l)

   1. Initial program 44.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 93.7

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 99.2%

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

Alternative 4: 97.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.9

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

5. Applied rewrites 91.9%

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

7. Applied rewrites 96.2%

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

9. Applied rewrites 97.7%

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

11. Applied rewrites 97.8%

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

Alternative 5: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 91.9

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

5. Applied rewrites 91.9%

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

7. Applied rewrites 96.2%

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

Alternative 6: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot k}{\ell} \cdot k\right) \cdot \sin k\right) \cdot \frac{k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{t \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.1e-8)
   (/ 2.0 (* (* (* (/ (* t k) l) k) (sin k)) (/ k (* l (cos k)))))
   (* (/ l k) (* (/ l k) (/ 2.0 (* t (* (sin k) (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.1e-8) {
		tmp = 2.0 / (((((t * k) / l) * k) * sin(k)) * (k / (l * cos(k))));
	} else {
		tmp = (l / k) * ((l / k) * (2.0 / (t * (sin(k) * tan(k)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.1d-8) then
        tmp = 2.0d0 / (((((t * k) / l) * k) * sin(k)) * (k / (l * cos(k))))
    else
        tmp = (l / k) * ((l / k) * (2.0d0 / (t * (sin(k) * tan(k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.1e-8) {
		tmp = 2.0 / (((((t * k) / l) * k) * Math.sin(k)) * (k / (l * Math.cos(k))));
	} else {
		tmp = (l / k) * ((l / k) * (2.0 / (t * (Math.sin(k) * Math.tan(k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.1e-8:
		tmp = 2.0 / (((((t * k) / l) * k) * math.sin(k)) * (k / (l * math.cos(k))))
	else:
		tmp = (l / k) * ((l / k) * (2.0 / (t * (math.sin(k) * math.tan(k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.1e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t * k) / l) * k) * sin(k)) * Float64(k / Float64(l * cos(k)))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(2.0 / Float64(t * Float64(sin(k) * tan(k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.1e-8)
		tmp = 2.0 / (((((t * k) / l) * k) * sin(k)) * (k / (l * cos(k))));
	else
		tmp = (l / k) * ((l / k) * (2.0 / (t * (sin(k) * tan(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.1e-8], N[(2.0 / N[(N[(N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.10000000000000032e-8

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 95.1%

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

   9. Applied rewrites 97.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 85.3%

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

   if 4.10000000000000032e-8 < k

   1. Initial program 39.1%

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

   3. Taylor expanded in t around 0

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 83.1%

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

   9. Applied rewrites 99.5%

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

Alternative 7: 82.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{t \cdot \left(\sin k \cdot \tan k\right)}}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.1e-8)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (* l (* (/ l k) (/ (/ 2.0 (* t (* (sin k) (tan k)))) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.1e-8) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = l * ((l / k) * ((2.0 / (t * (sin(k) * tan(k)))) / k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.1d-8) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = l * ((l / k) * ((2.0d0 / (t * (sin(k) * tan(k)))) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.1e-8) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = l * ((l / k) * ((2.0 / (t * (Math.sin(k) * Math.tan(k)))) / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.1e-8:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = l * ((l / k) * ((2.0 / (t * (math.sin(k) * math.tan(k)))) / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.1e-8)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(l * Float64(Float64(l / k) * Float64(Float64(2.0 / Float64(t * Float64(sin(k) * tan(k)))) / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.1e-8)
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	else
		tmp = l * ((l / k) * ((2.0 / (t * (sin(k) * tan(k)))) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.1e-8], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.10000000000000032e-8

   1. Initial program 36.0%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 77.5%

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

   11. Applied rewrites 82.6%

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

   if 4.10000000000000032e-8 < k

   1. Initial program 39.1%

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

   3. Taylor expanded in t around 0

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 83.1%

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

   9. Applied rewrites 93.9%

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

Alternative 8: 79.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{k}}{\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot k} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.5e-6)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (* (/ (/ (* l l) k) (* (* t (* (sin k) (tan k))) k)) 2.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e-6) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = (((l * l) / k) / ((t * (sin(k) * tan(k))) * k)) * 2.0;
	}
	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 <= 4.5d-6) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = (((l * l) / k) / ((t * (sin(k) * tan(k))) * k)) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e-6) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = (((l * l) / k) / ((t * (Math.sin(k) * Math.tan(k))) * k)) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.5e-6:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = (((l * l) / k) / ((t * (math.sin(k) * math.tan(k))) * k)) * 2.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.5e-6)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / k) / Float64(Float64(t * Float64(sin(k) * tan(k))) * k)) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.5e-6)
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	else
		tmp = (((l * l) / k) / ((t * (sin(k) * tan(k))) * k)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.5e-6], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision] / N[(N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000011e-6

   1. Initial program 36.0%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 77.5%

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

   11. Applied rewrites 82.6%

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

   if 4.50000000000000011e-6 < k

   1. Initial program 39.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 93.8

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

   5. Applied rewrites 93.8%

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

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

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

      2. lower-*.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 81.5

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

   8. Applied rewrites 81.5%

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

   10. Applied rewrites 84.3%

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

Alternative 9: 96.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in t around 0

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 78.0%

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

7. Applied rewrites 71.5%

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

9. Applied rewrites 96.0%

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

Alternative 10: 77.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(k \cdot k\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.5e-6)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (* (/ (* l l) (* (* t (* (sin k) (tan k))) (* k k))) 2.0)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e-6) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = ((l * l) / ((t * (sin(k) * tan(k))) * (k * k))) * 2.0;
	}
	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 <= 4.5d-6) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = ((l * l) / ((t * (sin(k) * tan(k))) * (k * k))) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.5e-6) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = ((l * l) / ((t * (Math.sin(k) * Math.tan(k))) * (k * k))) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.5e-6:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = ((l * l) / ((t * (math.sin(k) * math.tan(k))) * (k * k))) * 2.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.5e-6)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(l * l) / Float64(Float64(t * Float64(sin(k) * tan(k))) * Float64(k * k))) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.5e-6)
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	else
		tmp = ((l * l) / ((t * (sin(k) * tan(k))) * (k * k))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.5e-6], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / N[(N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000011e-6

   1. Initial program 36.0%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 77.5%

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

   9. Applied rewrites 77.5%

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

   11. Applied rewrites 82.6%

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

   if 4.50000000000000011e-6 < k

   1. Initial program 39.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 93.8

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

   5. Applied rewrites 93.8%

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

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

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

      2. lower-*.f64 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-sin.f64 81.5

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

   8. Applied rewrites 81.5%

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

   10. Applied rewrites 80.0%

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

Alternative 11: 73.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+167}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\ell \cdot \ell\right) \cdot 2\right) \cdot \cos k}{\left(k \cdot t\right) \cdot k}}{\mathsf{fma}\left(1, -0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1e+167)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (/ (/ (* (* (* l l) 2.0) (cos k)) (* (* k t) k)) (fma 1.0 -0.5 0.5))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e+167) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = ((((l * l) * 2.0) * cos(k)) / ((k * t) * k)) / fma(1.0, -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1e+167)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) * 2.0) * cos(k)) / Float64(Float64(k * t) * k)) / fma(1.0, -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1e+167], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(1.0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1e167

   1. Initial program 37.0%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 77.0%

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

   9. Applied rewrites 77.0%

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

   11. Applied rewrites 81.5%

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

   if 1e167 < l

   1. Initial program 35.2%

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

   3. Taylor expanded in t around 0

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   7. Applied rewrites 71.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.7%

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

   11. Taylor expanded in k around 0

       \[\leadsto \frac{\frac{\left(\left(\ell \cdot \ell\right) \cdot 2\right) \cdot \cos k}{\left(k \cdot t\right) \cdot k}}{\mathsf{fma}\left(1, \frac{-1}{2}, \frac{1}{2}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 70.7%

       \[\leadsto \frac{\frac{\left(\left(\ell \cdot \ell\right) \cdot 2\right) \cdot \cos k}{\left(k \cdot t\right) \cdot k}}{\mathsf{fma}\left(1, -0.5, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8800000000000:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot 2\right) \cdot \ell\right) \cdot \frac{\ell}{k}}{\left(k \cdot t\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8800000000000.0)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (/ (* (* (* (cos k) 2.0) l) (/ l k)) (* (* k t) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8800000000000.0) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = (((cos(k) * 2.0) * l) * (l / k)) / ((k * t) * (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8800000000000.0d0) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = (((cos(k) * 2.0d0) * l) * (l / k)) / ((k * t) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8800000000000.0) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = (((Math.cos(k) * 2.0) * l) * (l / k)) / ((k * t) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8800000000000.0:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = (((math.cos(k) * 2.0) * l) * (l / k)) / ((k * t) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8800000000000.0)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * 2.0) * l) * Float64(l / k)) / Float64(Float64(k * t) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8800000000000.0)
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	else
		tmp = (((cos(k) * 2.0) * l) * (l / k)) / ((k * t) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8800000000000.0], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * 2.0), $MachinePrecision] * l), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.8e12

   1. Initial program 35.9%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 77.1%

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

   11. Applied rewrites 82.1%

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

   if 8.8e12 < k

   1. Initial program 39.6%

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

   3. Taylor expanded in t around 0

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 63.6%

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

   10. Applied rewrites 64.0%

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

Alternative 13: 73.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+167}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\left(t \cdot k\right) \cdot k}}{k}}{k} \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot k} \cdot \frac{\ell \cdot \ell}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1e+167)
   (* (/ (/ (/ l (* (* t k) k)) k) k) (* 2.0 l))
   (* (/ (* 2.0 (cos k)) (* (* (* k k) t) k)) (/ (* l l) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e+167) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = ((2.0 * cos(k)) / (((k * k) * t) * 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 (l <= 1d+167) then
        tmp = (((l / ((t * k) * k)) / k) / k) * (2.0d0 * l)
    else
        tmp = ((2.0d0 * cos(k)) / (((k * k) * t) * k)) * ((l * l) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e+167) {
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	} else {
		tmp = ((2.0 * Math.cos(k)) / (((k * k) * t) * k)) * ((l * l) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1e+167:
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l)
	else:
		tmp = ((2.0 * math.cos(k)) / (((k * k) * t) * k)) * ((l * l) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1e+167)
		tmp = Float64(Float64(Float64(Float64(l / Float64(Float64(t * k) * k)) / k) / k) * Float64(2.0 * l));
	else
		tmp = Float64(Float64(Float64(2.0 * cos(k)) / Float64(Float64(Float64(k * k) * t) * k)) * Float64(Float64(l * l) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1e+167)
		tmp = (((l / ((t * k) * k)) / k) / k) * (2.0 * l);
	else
		tmp = ((2.0 * cos(k)) / (((k * k) * t) * k)) * ((l * l) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1e+167], N[(N[(N[(N[(l / N[(N[(t * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1e167

   1. Initial program 37.0%

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

   3. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 77.0%

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

   9. Applied rewrites 77.0%

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

   11. Applied rewrites 81.5%

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

   if 1e167 < l

   1. Initial program 35.2%

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

   3. Taylor expanded in t around 0

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 62.8%

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

Alternative 14: 72.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

11. Applied rewrites 77.2%

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

Alternative 15: 71.9% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

11. Applied rewrites 76.6%

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

Alternative 16: 72.8% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 76.3%

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

Alternative 17: 70.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

11. Applied rewrites 73.4%

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

Alternative 18: 70.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

Alternative 19: 70.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

9. Applied rewrites 73.3%

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

11. Applied rewrites 73.3%

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

Alternative 20: 70.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-*.f64 71.4

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

5. Applied rewrites 71.4%

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

7. Applied rewrites 73.3%

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

Reproduce

herbie shell --seed 2024305 
(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))))