Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.5% → 98.4%

Time: 13.8s

Alternatives: 14

Speedup: 9.6×

• 39.8% 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: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= (* l l) 5e-287)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (tan k) (sin k)) (/ (* (/ k l) t) (/ l k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-287) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((tan(k) * sin(k)) * (((k / l) * t) / (l / k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if ((l * l) <= 5d-287) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((tan(k) * sin(k)) * (((k / l) * t) / (l / k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-287) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((Math.tan(k) * Math.sin(k)) * (((k / l) * t) / (l / k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if (l * l) <= 5e-287:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((math.tan(k) * math.sin(k)) * (((k / l) * t) / (l / k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-287)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * sin(k)) * Float64(Float64(Float64(k / l) * t) / Float64(l / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if ((l * l) <= 5e-287)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((tan(k) * sin(k)) * (((k / l) * t) / (l / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-287], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000025e-287

   1. Initial program 32.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 64.8%

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

   11. Applied rewrites 99.7%

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

   if 5.00000000000000025e-287 < (*.f64 l l)

   1. Initial program 40.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot \frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-304}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= (* l l) 1e-304)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (/ (* k t) l) (/ k l)) (* (tan k) (sin k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 1e-304) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (tan(k) * sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if ((l * l) <= 1d-304) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((k * t) / l) * (k / l)) * (tan(k) * sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 1e-304) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (Math.tan(k) * Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if (l * l) <= 1e-304:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (math.tan(k) * math.sin(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-304)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) / l) * Float64(k / l)) * Float64(tan(k) * sin(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if ((l * l) <= 1e-304)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((k * t) / l) * (k / l)) * (tan(k) * sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-304], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999971e-305

   1. Initial program 30.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 66.4%

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

   9. Applied rewrites 64.5%

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

   11. Applied rewrites 99.8%

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

   if 9.99999999999999971e-305 < (*.f64 l l)

   1. Initial program 40.8%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 95.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-304}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 76000.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (if (<= k 1.9e+245)
       (/ 2.0 (* (/ (* (* k t) k) (* l l)) (* (tan k) (sin k))))
       (/ 2.0 (* (* (/ k l) (/ k l)) (* (pow (sin k) 2.0) t)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k <= 1.9e+245) {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * (pow(sin(k), 2.0) * 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) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 76000.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else if (k <= 1.9d+245) then
        tmp = 2.0d0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)))
    else
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((sin(k) ** 2.0d0) * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k <= 1.9e+245) {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * (Math.pow(Math.sin(k), 2.0) * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 76000.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	elif k <= 1.9e+245:
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (math.tan(k) * math.sin(k)))
	else:
		tmp = 2.0 / (((k / l) * (k / l)) * (math.pow(math.sin(k), 2.0) * t))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 76000.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	elseif (k <= 1.9e+245)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) * k) / Float64(l * l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64((sin(k) ^ 2.0) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 76000.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	elseif (k <= 1.9e+245)
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	else
		tmp = 2.0 / (((k / l) * (k / l)) * ((sin(k) ^ 2.0) * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 76000.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.9e+245], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 76000

   1. Initial program 38.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 67.5%

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

   11. Applied rewrites 84.1%

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

   if 76000 < k < 1.9e245

   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.3%

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

   11. Applied rewrites 73.9%

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

   if 1.9e245 < k

   1. Initial program 45.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 67.9%

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

   10. Applied rewrites 69.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left({\sin k}^{2} \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 76000.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (if (<= k 1.9e+245)
       (/ 2.0 (* (/ (* (* k t) k) (* l l)) (* (tan k) (sin k))))
       (/ 2.0 (* (* (* (pow (sin k) 2.0) (/ k l)) t) (/ k l)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k <= 1.9e+245) {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / (((pow(sin(k), 2.0) * (k / l)) * t) * (k / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 76000.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else if (k <= 1.9d+245) then
        tmp = 2.0d0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)))
    else
        tmp = 2.0d0 / ((((sin(k) ** 2.0d0) * (k / l)) * t) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k <= 1.9e+245) {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) * (k / l)) * t) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 76000.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	elif k <= 1.9e+245:
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (math.tan(k) * math.sin(k)))
	else:
		tmp = 2.0 / (((math.pow(math.sin(k), 2.0) * (k / l)) * t) * (k / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 76000.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	elseif (k <= 1.9e+245)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) * k) / Float64(l * l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) * Float64(k / l)) * t) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 76000.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	elseif (k <= 1.9e+245)
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	else
		tmp = 2.0 / ((((sin(k) ^ 2.0) * (k / l)) * t) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 76000.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.9e+245], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 76000

   1. Initial program 38.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 67.5%

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

   11. Applied rewrites 84.1%

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

   if 76000 < k < 1.9e245

   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.3%

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

   11. Applied rewrites 73.9%

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

   if 1.9e245 < k

   1. Initial program 45.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 68.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot t\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\ell} \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 76000.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (if (<= k 1.9e+245)
       (/ 2.0 (* (/ (* (* k t) k) (* l l)) (* (tan k) (sin k))))
       (/ 2.0 (* (* (/ (* (pow (sin k) 2.0) t) l) k) (/ k l)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k <= 1.9e+245) {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	} else {
		tmp = 2.0 / ((((pow(sin(k), 2.0) * t) / l) * k) * (k / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 76000.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else if (k <= 1.9d+245) then
        tmp = 2.0d0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)))
    else
        tmp = 2.0d0 / (((((sin(k) ** 2.0d0) * t) / l) * k) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else if (k <= 1.9e+245) {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (Math.tan(k) * Math.sin(k)));
	} else {
		tmp = 2.0 / ((((Math.pow(Math.sin(k), 2.0) * t) / l) * k) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 76000.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	elif k <= 1.9e+245:
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (math.tan(k) * math.sin(k)))
	else:
		tmp = 2.0 / ((((math.pow(math.sin(k), 2.0) * t) / l) * k) * (k / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 76000.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	elseif (k <= 1.9e+245)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) * k) / Float64(l * l)) * Float64(tan(k) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t) / l) * k) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 76000.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	elseif (k <= 1.9e+245)
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	else
		tmp = 2.0 / (((((sin(k) ^ 2.0) * t) / l) * k) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 76000.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.9e+245], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 76000

   1. Initial program 38.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 67.5%

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

   11. Applied rewrites 84.1%

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

   if 76000 < k < 1.9e245

   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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.3%

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

   11. Applied rewrites 73.9%

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

   if 1.9e245 < k

   1. Initial program 45.0%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 67.9%

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

   10. Applied rewrites 68.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+245}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\sin k}^{2} \cdot t}{\ell} \cdot k\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 5e-17)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (/ k l) t) (* (tan k) (sin k))) (/ k l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 5e-17) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k / l) * t) * (tan(k) * sin(k))) * (k / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 5d-17) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((k / l) * t) * (tan(k) * sin(k))) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 5e-17) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k / l) * t) * (Math.tan(k) * Math.sin(k))) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 5e-17:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((k / l) * t) * (math.tan(k) * math.sin(k))) * (k / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 5e-17)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * t) * Float64(tan(k) * sin(k))) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 5e-17)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((k / l) * t) * (tan(k) * sin(k))) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 5e-17], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999999e-17

   1. Initial program 38.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 68.4%

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

   11. Applied rewrites 84.4%

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

   if 4.9999999999999999e-17 < k

   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 t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

   11. Applied rewrites 98.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 5e-19)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (* (/ t l) (/ k l)) (* (tan k) (sin k))) k)))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 5e-19) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((t / l) * (k / l)) * (tan(k) * sin(k))) * k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 5d-19) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((t / l) * (k / l)) * (tan(k) * sin(k))) * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 5e-19) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((t / l) * (k / l)) * (Math.tan(k) * Math.sin(k))) * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 5e-19:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((t / l) * (k / l)) * (math.tan(k) * math.sin(k))) * k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 5e-19)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(k / l)) * Float64(tan(k) * sin(k))) * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 5e-19)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((t / l) * (k / l)) * (tan(k) * sin(k))) * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 5e-19], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000004e-19

   1. Initial program 38.7%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 72.0%

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

   9. Applied rewrites 68.4%

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

   11. Applied rewrites 84.4%

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

   if 5.0000000000000004e-19 < k

   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 t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 90.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 76000.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (/ (* (* k t) k) (* l l)) (* (tan k) (sin k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 76000.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 76000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (Math.tan(k) * Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 76000.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (math.tan(k) * math.sin(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 76000.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) * k) / Float64(l * l)) * Float64(tan(k) * sin(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 76000.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((((k * t) * k) / (l * l)) * (tan(k) * sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 76000.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 76000

   1. Initial program 38.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 67.5%

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

   11. Applied rewrites 84.1%

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

   if 76000 < k

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

   11. Applied rewrites 68.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 76000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 4.8:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 4.8)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (/ (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t) k) l) (/ k l))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 4.8) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((0.5 - (cos((k + k)) * 0.5)) * t) * k) / l) * (k / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 4.8d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((0.5d0 - (cos((k + k)) * 0.5d0)) * t) * k) / l) * (k / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 4.8) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((0.5 - (Math.cos((k + k)) * 0.5)) * t) * k) / l) * (k / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 4.8:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((((0.5 - (math.cos((k + k)) * 0.5)) * t) * k) / l) * (k / l))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 4.8)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t) * k) / l) * Float64(k / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 4.8)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((0.5 - (cos((k + k)) * 0.5)) * t) * k) / l) * (k / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 4.8], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.79999999999999982

   1. Initial program 38.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 73.7

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 71.4%

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

   9. Applied rewrites 67.8%

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

   11. Applied rewrites 84.5%

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

   if 4.79999999999999982 < k

   1. Initial program 37.9%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.6%

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

   10. Applied rewrites 60.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\right) \cdot k}{\ell} \cdot \frac{k}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;k \leq 3050000000:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot k\right) \cdot \frac{k}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= k 3050000000.0)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* (/ (* (* k k) t) (* l l)) k) (/ k (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 3050000000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k * k) * t) / (l * l)) * k) * (k / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if (k <= 3050000000.0d0) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / (((((k * k) * t) / (l * l)) * k) * (k / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if (k <= 3050000000.0) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / (((((k * k) * t) / (l * l)) * k) * (k / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if k <= 3050000000.0:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / (((((k * k) * t) / (l * l)) * k) * (k / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (k <= 3050000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k * k) * t) / Float64(l * l)) * k) * Float64(k / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if (k <= 3050000000.0)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / (((((k * k) * t) / (l * l)) * k) * (k / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[k, 3050000000.0], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.05e9

   1. Initial program 37.9%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 67.2%

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

   11. Applied rewrites 83.7%

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

   if 3.05e9 < k

   1. Initial program 38.7%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 74.9%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 55.4%

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

   12. Applied rewrites 55.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3050000000:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell} \cdot k\right) \cdot \frac{k}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot k\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot t\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) k)))
   (if (<= (* l l) 5e-287)
     (/ 2.0 (* (* t_1 t) t_1))
     (/ 2.0 (* (* k k) (/ (* (/ k l) t) (/ l k)))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-287) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((k * k) * (((k / l) * t) / (l / k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * k
    if ((l * l) <= 5d-287) then
        tmp = 2.0d0 / ((t_1 * t) * t_1)
    else
        tmp = 2.0d0 / ((k * k) * (((k / l) * t) / (l / k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * k;
	double tmp;
	if ((l * l) <= 5e-287) {
		tmp = 2.0 / ((t_1 * t) * t_1);
	} else {
		tmp = 2.0 / ((k * k) * (((k / l) * t) / (l / k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * k
	tmp = 0
	if (l * l) <= 5e-287:
		tmp = 2.0 / ((t_1 * t) * t_1)
	else:
		tmp = 2.0 / ((k * k) * (((k / l) * t) / (l / k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-287)
		tmp = Float64(2.0 / Float64(Float64(t_1 * t) * t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(k / l) * t) / Float64(l / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * k;
	tmp = 0.0;
	if ((l * l) <= 5e-287)
		tmp = 2.0 / ((t_1 * t) * t_1);
	else
		tmp = 2.0 / ((k * k) * (((k / l) * t) / (l / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-287], N[(2.0 / N[(N[(t$95$1 * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000025e-287

   1. Initial program 32.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 64.8%

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

   11. Applied rewrites 99.7%

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

   if 5.00000000000000025e-287 < (*.f64 l l)

   1. Initial program 40.5%

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

   3. Taylor expanded in t around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 68.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k}{\ell} \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 67.8

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

5. Applied rewrites 67.8%

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

7. Applied rewrites 66.0%

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

9. Applied rewrites 62.7%

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

11. Applied rewrites 75.6%

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

12. Final simplification 75.6%

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

Alternative 13: 72.4% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 93.0%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 75.0%

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

9. Taylor expanded in k around 0

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

11. Applied rewrites 71.9%

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

12. Final simplification 71.9%

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

Alternative 14: 64.5% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 67.8

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

5. Applied rewrites 67.8%

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

7. Applied rewrites 66.0%

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

8. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2024270 
(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))))