Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 95.0%

Time: 16.8s

Alternatives: 16

Speedup: 9.6×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.5% 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: 95.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\ell}{\tan k}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ l (tan k)) (* (* k (/ (sin k) l)) (* (* k t) 0.5))))

C

double code(double t, double l, double k) {
	return (l / tan(k)) / ((k * (sin(k) / l)) * ((k * t) * 0.5));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / tan(k)) / ((k * (sin(k) / l)) * ((k * t) * 0.5d0))
end function

Java

public static double code(double t, double l, double k) {
	return (l / Math.tan(k)) / ((k * (Math.sin(k) / l)) * ((k * t) * 0.5));
}

Python

def code(t, l, k):
	return (l / math.tan(k)) / ((k * (math.sin(k) / l)) * ((k * t) * 0.5))

Julia

function code(t, l, k)
	return Float64(Float64(l / tan(k)) / Float64(Float64(k * Float64(sin(k) / l)) * Float64(Float64(k * t) * 0.5)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / tan(k)) / ((k * (sin(k) / l)) * ((k * t) * 0.5));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

4. Applied egg-rr 30.2%

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

6. Applied egg-rr 87.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. *-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. clear-num N/A

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

   15. div-inv N/A

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

   16. lower-/.f64 N/A

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

8. Applied egg-rr 92.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]
9. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 96.9

       \[\leadsto \frac{\frac{\ell}{\tan k}}{\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot 0.5\right)}} \]

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 96.9

       \[\leadsto \frac{\frac{\ell}{\tan k}}{\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot 0.5\right)} \]

10. Applied egg-rr 96.9%

    \[\leadsto \frac{\frac{\ell}{\tan k}}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)}} \]

11. Final simplification 96.9%

    \[\leadsto \frac{\frac{\ell}{\tan k}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)} \]
12. Add Preprocessing

Alternative 2: 86.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot t\_1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 8 \cdot 10^{+279}:\\ \;\;\;\;\frac{2}{\sin k \cdot t\_1} \cdot \left(\ell \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \left(\sin k \cdot \frac{0.5 \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= (* l l) 5e-167)
     (/ (/ l k) (* (/ (sin k) l) (* 0.5 t_1)))
     (if (<= (* l l) 8e+279)
       (* (/ 2.0 (* (sin k) t_1)) (* l (/ l (tan k))))
       (/ l (* (tan k) (* (sin k) (/ (* 0.5 (* t (* k k))) l))))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if ((l * l) <= 5e-167) {
		tmp = (l / k) / ((sin(k) / l) * (0.5 * t_1));
	} else if ((l * l) <= 8e+279) {
		tmp = (2.0 / (sin(k) * t_1)) * (l * (l / tan(k)));
	} else {
		tmp = l / (tan(k) * (sin(k) * ((0.5 * (t * (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 * (k * t)
    if ((l * l) <= 5d-167) then
        tmp = (l / k) / ((sin(k) / l) * (0.5d0 * t_1))
    else if ((l * l) <= 8d+279) then
        tmp = (2.0d0 / (sin(k) * t_1)) * (l * (l / tan(k)))
    else
        tmp = l / (tan(k) * (sin(k) * ((0.5d0 * (t * (k * k))) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if ((l * l) <= 5e-167) {
		tmp = (l / k) / ((Math.sin(k) / l) * (0.5 * t_1));
	} else if ((l * l) <= 8e+279) {
		tmp = (2.0 / (Math.sin(k) * t_1)) * (l * (l / Math.tan(k)));
	} else {
		tmp = l / (Math.tan(k) * (Math.sin(k) * ((0.5 * (t * (k * k))) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if (l * l) <= 5e-167:
		tmp = (l / k) / ((math.sin(k) / l) * (0.5 * t_1))
	elif (l * l) <= 8e+279:
		tmp = (2.0 / (math.sin(k) * t_1)) * (l * (l / math.tan(k)))
	else:
		tmp = l / (math.tan(k) * (math.sin(k) * ((0.5 * (t * (k * k))) / l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (Float64(l * l) <= 5e-167)
		tmp = Float64(Float64(l / k) / Float64(Float64(sin(k) / l) * Float64(0.5 * t_1)));
	elseif (Float64(l * l) <= 8e+279)
		tmp = Float64(Float64(2.0 / Float64(sin(k) * t_1)) * Float64(l * Float64(l / tan(k))));
	else
		tmp = Float64(l / Float64(tan(k) * Float64(sin(k) * Float64(Float64(0.5 * Float64(t * Float64(k * k))) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if ((l * l) <= 5e-167)
		tmp = (l / k) / ((sin(k) / l) * (0.5 * t_1));
	elseif ((l * l) <= 8e+279)
		tmp = (2.0 / (sin(k) * t_1)) * (l * (l / tan(k)));
	else
		tmp = l / (tan(k) * (sin(k) * ((0.5 * (t * (k * k))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-167], N[(N[(l / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 8e+279], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(0.5 * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.0000000000000002e-167

   1. Initial program 25.7%

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

   4. Applied egg-rr 26.8%

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

   6. Applied egg-rr 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 94.3

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   11. Simplified 94.3%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   if 5.0000000000000002e-167 < (*.f64 l l) < 8.00000000000000046e279

   1. Initial program 54.5%

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

   4. Applied egg-rr 36.8%

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

   6. Applied egg-rr 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied egg-rr 96.7%

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

   if 8.00000000000000046e279 < (*.f64 l l)

   1. Initial program 40.6%

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

   4. Applied egg-rr 26.1%

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

   6. Applied egg-rr 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]
   9. Step-by-step derivation

      1. lift-tan.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 83.3

          \[\leadsto \frac{\ell}{\color{blue}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)\right)}} \]

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 83.3

          \[\leadsto \frac{\ell}{\tan k \cdot \left(\sin k \cdot \color{blue}{\frac{\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5}{\ell}}\right)} \]

   10. Applied egg-rr 79.1%

       \[\leadsto \color{blue}{\frac{\ell}{\tan k \cdot \left(\sin k \cdot \frac{0.5 \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 8 \cdot 10^{+279}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k \cdot \left(\sin k \cdot \frac{0.5 \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot t\_1} \cdot \left(\ell \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 4.5e-6)
     (/ (/ l k) (* (/ (sin k) l) (* 0.5 t_1)))
     (* (/ 2.0 (* (sin k) t_1)) (* l (/ l (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 4.5e-6) {
		tmp = (l / k) / ((sin(k) / l) * (0.5 * t_1));
	} else {
		tmp = (2.0 / (sin(k) * t_1)) * (l * (l / tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k * t)
    if (k <= 4.5d-6) then
        tmp = (l / k) / ((sin(k) / l) * (0.5d0 * t_1))
    else
        tmp = (2.0d0 / (sin(k) * t_1)) * (l * (l / tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 4.5e-6) {
		tmp = (l / k) / ((Math.sin(k) / l) * (0.5 * t_1));
	} else {
		tmp = (2.0 / (Math.sin(k) * t_1)) * (l * (l / Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 4.5e-6:
		tmp = (l / k) / ((math.sin(k) / l) * (0.5 * t_1))
	else:
		tmp = (2.0 / (math.sin(k) * t_1)) * (l * (l / math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 4.5e-6)
		tmp = Float64(Float64(l / k) / Float64(Float64(sin(k) / l) * Float64(0.5 * t_1)));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k) * t_1)) * Float64(l * Float64(l / tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 4.5e-6)
		tmp = (l / k) / ((sin(k) / l) * (0.5 * t_1));
	else
		tmp = (2.0 / (sin(k) * t_1)) * (l * (l / tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.5e-6], N[(N[(l / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000011e-6

   1. Initial program 40.4%

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

   4. Applied egg-rr 30.0%

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

   6. Applied egg-rr 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 83.0

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   11. Simplified 83.0%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   if 4.50000000000000011e-6 < k

   1. Initial program 39.2%

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

   4. Applied egg-rr 31.2%

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

   6. Applied egg-rr 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   8. Applied egg-rr 77.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\ell}{\tan k}}{k \cdot \left(\frac{\sin k}{\ell} \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ l (tan k)) (* k (* (/ (sin k) l) (* (* k t) 0.5)))))

C

double code(double t, double l, double k) {
	return (l / tan(k)) / (k * ((sin(k) / l) * ((k * t) * 0.5)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / tan(k)) / (k * ((sin(k) / l) * ((k * t) * 0.5d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l / Math.tan(k)) / (k * ((Math.sin(k) / l) * ((k * t) * 0.5)));
}

Python

def code(t, l, k):
	return (l / math.tan(k)) / (k * ((math.sin(k) / l) * ((k * t) * 0.5)))

Julia

function code(t, l, k)
	return Float64(Float64(l / tan(k)) / Float64(k * Float64(Float64(sin(k) / l) * Float64(Float64(k * t) * 0.5))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / tan(k)) / (k * ((sin(k) / l) * ((k * t) * 0.5)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

4. Applied egg-rr 30.2%

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

6. Applied egg-rr 87.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. *-commutative N/A

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

   13. lift-/.f64 N/A

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

   14. clear-num N/A

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

   15. div-inv N/A

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

   16. lower-/.f64 N/A

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

8. Applied egg-rr 92.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]
9. Step-by-step derivation

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 96.8

       \[\leadsto \frac{\frac{\ell}{\tan k}}{k \cdot \left(\color{blue}{\left(\left(t \cdot k\right) \cdot 0.5\right)} \cdot \frac{\sin k}{\ell}\right)} \]

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 96.8

       \[\leadsto \frac{\frac{\ell}{\tan k}}{k \cdot \left(\left(\color{blue}{\left(k \cdot t\right)} \cdot 0.5\right) \cdot \frac{\sin k}{\ell}\right)} \]

10. Applied egg-rr 96.8%

    \[\leadsto \frac{\frac{\ell}{\tan k}}{\color{blue}{k \cdot \left(\left(\left(k \cdot t\right) \cdot 0.5\right) \cdot \frac{\sin k}{\ell}\right)}} \]

11. Final simplification 96.8%

    \[\leadsto \frac{\frac{\ell}{\tan k}}{k \cdot \left(\frac{\sin k}{\ell} \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)\right)} \]
12. Add Preprocessing

Alternative 5: 79.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-6)
   (/ (/ l k) (* (/ (sin k) l) (* 0.5 (* k (* k t)))))
   (* (* l l) (/ 2.0 (* (sin k) (* (tan k) (* t (* k k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-6) {
		tmp = (l / k) / ((sin(k) / l) * (0.5 * (k * (k * t))));
	} else {
		tmp = (l * l) * (2.0 / (sin(k) * (tan(k) * (t * (k * k)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.5d-6) then
        tmp = (l / k) / ((sin(k) / l) * (0.5d0 * (k * (k * t))))
    else
        tmp = (l * l) * (2.0d0 / (sin(k) * (tan(k) * (t * (k * k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-6) {
		tmp = (l / k) / ((Math.sin(k) / l) * (0.5 * (k * (k * t))));
	} else {
		tmp = (l * l) * (2.0 / (Math.sin(k) * (Math.tan(k) * (t * (k * k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.5e-6:
		tmp = (l / k) / ((math.sin(k) / l) * (0.5 * (k * (k * t))))
	else:
		tmp = (l * l) * (2.0 / (math.sin(k) * (math.tan(k) * (t * (k * k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-6)
		tmp = Float64(Float64(l / k) / Float64(Float64(sin(k) / l) * Float64(0.5 * Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t * Float64(k * k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-6)
		tmp = (l / k) / ((sin(k) / l) * (0.5 * (k * (k * t))));
	else
		tmp = (l * l) * (2.0 / (sin(k) * (tan(k) * (t * (k * k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.5e-6], N[(N[(l / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5000000000000002e-6

   1. Initial program 40.4%

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

   4. Applied egg-rr 30.0%

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

   6. Applied egg-rr 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 83.0

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   11. Simplified 83.0%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   if 2.5000000000000002e-6 < k

   1. Initial program 39.2%

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

   4. Applied egg-rr 31.2%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 76.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\sin k \cdot \left(\tan k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ l (* (tan k) (* (/ (sin k) l) (* 0.5 (* k (* k t)))))))

C

double code(double t, double l, double k) {
	return l / (tan(k) * ((sin(k) / l) * (0.5 * (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 = l / (tan(k) * ((sin(k) / l) * (0.5d0 * (k * (k * t)))))
end function

Java

public static double code(double t, double l, double k) {
	return l / (Math.tan(k) * ((Math.sin(k) / l) * (0.5 * (k * (k * t)))));
}

Python

def code(t, l, k):
	return l / (math.tan(k) * ((math.sin(k) / l) * (0.5 * (k * (k * t)))))

Julia

function code(t, l, k)
	return Float64(l / Float64(tan(k) * Float64(Float64(sin(k) / l) * Float64(0.5 * Float64(k * Float64(k * t))))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l / (tan(k) * ((sin(k) / l) * (0.5 * (k * (k * t)))));
end

Wolfram

code[t_, l_, k_] := N[(l / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

4. Applied egg-rr 30.2%

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

6. Applied egg-rr 87.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. associate-/r/ N/A

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

   9. *-commutative N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

   12. lower-/.f64 N/A

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

   13. div-inv N/A

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

   14. lift-/.f64 N/A

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

8. Applied egg-rr 91.2%

   \[\leadsto \color{blue}{\frac{\ell}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)\right)}} \]

9. Final simplification 91.2%

   \[\leadsto \frac{\ell}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right)} \]
10. Add Preprocessing

Alternative 7: 67.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 17000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.3333333333333333, \ell\right)}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 17000.0)
   (/
    (/ (fma l (* (* k k) -0.3333333333333333) l) k)
    (* (/ (sin k) l) (* 0.5 (* k (* k t)))))
   (/ (* 2.0 (* l (* l (cos k)))) (* k (* k (* t (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 17000.0) {
		tmp = (fma(l, ((k * k) * -0.3333333333333333), l) / k) / ((sin(k) / l) * (0.5 * (k * (k * t))));
	} else {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 17000.0)
		tmp = Float64(Float64(fma(l, Float64(Float64(k * k) * -0.3333333333333333), l) / k) / Float64(Float64(sin(k) / l) * Float64(0.5 * Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(k * Float64(t * Float64(k * k)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 17000.0], N[(N[(N[(l * N[(N[(k * k), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + l), $MachinePrecision] / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 17000

   1. Initial program 40.7%

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

   4. Applied egg-rr 30.3%

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

   6. Applied egg-rr 88.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\ell, \color{blue}{\left(k \cdot k\right)} \cdot \frac{-1}{3}, \ell\right)}{k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot \frac{1}{2}\right)} \]

       9. lower-*.f64 74.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\ell, \color{blue}{\left(k \cdot k\right)} \cdot -0.3333333333333333, \ell\right)}{k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   11. Simplified 74.3%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.3333333333333333, \ell\right)}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   if 17000 < k

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 76.6

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

   5. Simplified 76.6%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 65.6

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

   8. Simplified 65.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 17000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\ell, \left(k \cdot k\right) \cdot -0.3333333333333333, \ell\right)}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+192}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+192)
   (/ (/ l k) (* (/ (sin k) l) (* 0.5 (* k (* k t)))))
   (/ (* 2.0 (* l (* l (cos k)))) (* k (* k (* t (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+192) {
		tmp = (l / k) / ((sin(k) / l) * (0.5 * (k * (k * t))));
	} else {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+192) then
        tmp = (l / k) / ((sin(k) / l) * (0.5d0 * (k * (k * t))))
    else
        tmp = (2.0d0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+192) {
		tmp = (l / k) / ((Math.sin(k) / l) * (0.5 * (k * (k * t))));
	} else {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (k * (k * (t * (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+192:
		tmp = (l / k) / ((math.sin(k) / l) * (0.5 * (k * (k * t))))
	else:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (k * (k * (t * (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+192)
		tmp = Float64(Float64(l / k) / Float64(Float64(sin(k) / l) * Float64(0.5 * Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(k * Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+192)
		tmp = (l / k) / ((sin(k) / l) * (0.5 * (k * (k * t))));
	else
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+192], N[(N[(l / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(0.5 * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000004e192

   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

   4. Applied egg-rr 28.6%

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

   6. Applied egg-rr 91.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. *-commutative N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)}} \]

   9. Taylor expanded in k around 0

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

       1. lower-/.f64 85.5

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   11. Simplified 85.5%

       \[\leadsto \frac{\color{blue}{\frac{\ell}{k}}}{\frac{\sin k}{\ell} \cdot \left(\left(k \cdot \left(t \cdot k\right)\right) \cdot 0.5\right)} \]

   if 1.00000000000000004e192 < (*.f64 l l)

   1. Initial program 48.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 70.7

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

   8. Simplified 70.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+192}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell} \cdot \left(0.5 \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-190}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{1}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-190)
   (* (* l 2.0) (/ 1.0 (/ (* k (* k t)) (/ (/ l k) k))))
   (/ (* 2.0 (* l (* l (cos k)))) (* k (* k (* t (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-190) {
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)));
	} else {
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-190) then
        tmp = (l * 2.0d0) * (1.0d0 / ((k * (k * t)) / ((l / k) / k)))
    else
        tmp = (2.0d0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-190) {
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)));
	} else {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (k * (k * (t * (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-190:
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)))
	else:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (k * (k * (t * (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-190)
		tmp = Float64(Float64(l * 2.0) * Float64(1.0 / Float64(Float64(k * Float64(k * t)) / Float64(Float64(l / k) / k))));
	else
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64(k * Float64(k * Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-190)
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)));
	else
		tmp = (2.0 * (l * (l * cos(k)))) / (k * (k * (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-190], N[(N[(l * 2.0), $MachinePrecision] * N[(1.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-190

   1. Initial program 23.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 53.1

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

   5. Simplified 53.1%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 78.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 82.1

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

   7. Applied egg-rr 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t \cdot \left(k \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t \cdot \left(k \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\frac{\frac{\ell}{k}}{k}}} \]

      17. *-commutative N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      19. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      20. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k \cdot \left(t \cdot k\right)}{\color{blue}{\frac{\frac{\ell}{k}}{k}}}} \]

      21. lower-/.f64 89.7

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\color{blue}{\frac{\ell}{k}}}{k}}} \]

   9. Applied egg-rr 89.7%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

   if 1e-190 < (*.f64 l l)

   1. Initial program 48.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 85.9

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

   5. Simplified 85.9%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 73.2

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

   8. Simplified 73.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-190}:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{1}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{1}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 175000000000.0)
   (* (* l 2.0) (/ 1.0 (/ (* k (* k t)) (/ (/ l k) k))))
   (* (/ l k) (/ (* l -0.3333333333333333) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 175000000000.0d0) then
        tmp = (l * 2.0d0) * (1.0d0 / ((k * (k * t)) / ((l / k) / k)))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 175000000000.0:
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 175000000000.0)
		tmp = Float64(Float64(l * 2.0) * Float64(1.0 / Float64(Float64(k * Float64(k * t)) / Float64(Float64(l / k) / k))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 175000000000.0)
		tmp = (l * 2.0) * (1.0 / ((k * (k * t)) / ((l / k) / k)));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 175000000000.0], N[(N[(l * 2.0), $MachinePrecision] * N[(1.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e11

   1. Initial program 40.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.4

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

   5. Simplified 64.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 77.2

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

   7. Applied egg-rr 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. clear-num N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t \cdot \left(k \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t \cdot \left(k \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(k \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{t \cdot \color{blue}{\left(k \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      16. associate-*r* N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{\left(t \cdot k\right) \cdot k}}{\frac{\frac{\ell}{k}}{k}}} \]

      17. *-commutative N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{\color{blue}{k \cdot \left(t \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      19. lower-*.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k \cdot \color{blue}{\left(t \cdot k\right)}}{\frac{\frac{\ell}{k}}{k}}} \]

      20. lower-/.f64 N/A

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k \cdot \left(t \cdot k\right)}{\color{blue}{\frac{\frac{\ell}{k}}{k}}}} \]

      21. lower-/.f64 80.8

          \[\leadsto \left(2 \cdot \ell\right) \cdot \frac{1}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\color{blue}{\frac{\ell}{k}}}{k}}} \]

   9. Applied egg-rr 80.8%

      \[\leadsto \left(2 \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\frac{\ell}{k}}{k}}}} \]

   if 1.75e11 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 63.7

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   11. Simplified 63.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       9. lower-*.f64 67.4

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot -0.3333333333333333}}{t \cdot k} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{t \cdot k}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

       12. lower-*.f64 67.4

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]

   13. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{1}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\frac{\ell \cdot 2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 175000000000.0)
   (* (/ (* l 2.0) (* t (* k k))) (/ l (* k k)))
   (* (/ l k) (/ (* l -0.3333333333333333) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = ((l * 2.0) / (t * (k * k))) * (l / (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 175000000000.0d0) then
        tmp = ((l * 2.0d0) / (t * (k * k))) * (l / (k * k))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = ((l * 2.0) / (t * (k * k))) * (l / (k * k));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 175000000000.0:
		tmp = ((l * 2.0) / (t * (k * k))) * (l / (k * k))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 175000000000.0)
		tmp = Float64(Float64(Float64(l * 2.0) / Float64(t * Float64(k * k))) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 175000000000.0)
		tmp = ((l * 2.0) / (t * (k * k))) * (l / (k * k));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 175000000000.0], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e11

   1. Initial program 40.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.4

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

   5. Simplified 64.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 80.5

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

   7. Applied egg-rr 80.5%

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

   if 1.75e11 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 63.7

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   11. Simplified 63.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       9. lower-*.f64 67.4

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot -0.3333333333333333}}{t \cdot k} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{t \cdot k}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

       12. lower-*.f64 67.4

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]

   13. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\frac{\ell \cdot 2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 175000000000.0)
   (* (* l 2.0) (/ (/ l k) (* k (* k (* k t)))))
   (* (/ l k) (/ (* l -0.3333333333333333) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * ((l / k) / (k * (k * (k * t))));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 175000000000.0d0) then
        tmp = (l * 2.0d0) * ((l / k) / (k * (k * (k * t))))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * ((l / k) / (k * (k * (k * t))));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 175000000000.0:
		tmp = (l * 2.0) * ((l / k) / (k * (k * (k * t))))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 175000000000.0)
		tmp = Float64(Float64(l * 2.0) * Float64(Float64(l / k) / Float64(k * Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 175000000000.0)
		tmp = (l * 2.0) * ((l / k) / (k * (k * (k * t))));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 175000000000.0], N[(N[(l * 2.0), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e11

   1. Initial program 40.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.4

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

   5. Simplified 64.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 77.2

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

   7. Applied egg-rr 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 77.5

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 78.7

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 79.2

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

   9. Applied egg-rr 79.2%

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

   if 1.75e11 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 63.7

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   11. Simplified 63.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       9. lower-*.f64 67.4

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot -0.3333333333333333}}{t \cdot k} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{t \cdot k}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

       12. lower-*.f64 67.4

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]

   13. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 175000000000.0)
   (* (* l 2.0) (/ l (* k (* (* k t) (* k k)))))
   (* (/ l k) (/ (* l -0.3333333333333333) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * (l / (k * ((k * t) * (k * k))));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 175000000000.0d0) then
        tmp = (l * 2.0d0) * (l / (k * ((k * t) * (k * k))))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * (l / (k * ((k * t) * (k * k))));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 175000000000.0:
		tmp = (l * 2.0) * (l / (k * ((k * t) * (k * k))))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 175000000000.0)
		tmp = Float64(Float64(l * 2.0) * Float64(l / Float64(k * Float64(Float64(k * t) * Float64(k * k)))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 175000000000.0)
		tmp = (l * 2.0) * (l / (k * ((k * t) * (k * k))));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 175000000000.0], N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(k * N[(N[(k * t), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e11

   1. Initial program 40.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.4

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

   5. Simplified 64.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 77.2

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

   7. Applied egg-rr 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 78.5

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

   9. Applied egg-rr 78.5%

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

   if 1.75e11 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 63.7

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   11. Simplified 63.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       9. lower-*.f64 67.4

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot -0.3333333333333333}}{t \cdot k} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{t \cdot k}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

       12. lower-*.f64 67.4

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]

   13. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 175000000000.0)
   (* (* l 2.0) (/ l (* k (* t (* k (* k k))))))
   (* (/ l k) (/ (* l -0.3333333333333333) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * (l / (k * (t * (k * (k * k)))));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 175000000000.0d0) then
        tmp = (l * 2.0d0) * (l / (k * (t * (k * (k * k)))))
    else
        tmp = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 175000000000.0) {
		tmp = (l * 2.0) * (l / (k * (t * (k * (k * k)))));
	} else {
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 175000000000.0:
		tmp = (l * 2.0) * (l / (k * (t * (k * (k * k)))))
	else:
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 175000000000.0)
		tmp = Float64(Float64(l * 2.0) * Float64(l / Float64(k * Float64(t * Float64(k * Float64(k * k))))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 175000000000.0)
		tmp = (l * 2.0) * (l / (k * (t * (k * (k * k)))));
	else
		tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 175000000000.0], N[(N[(l * 2.0), $MachinePrecision] * N[(l / N[(k * N[(t * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e11

   1. Initial program 40.7%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 64.4

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

   5. Simplified 64.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 74.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 77.2

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

   7. Applied egg-rr 77.2%

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

   if 1.75e11 < 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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-sin.f64 74.3

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

   5. Simplified 74.3%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 63.7

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   11. Simplified 63.7%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       7. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

       9. lower-*.f64 67.4

          \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot -0.3333333333333333}}{t \cdot k} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{t \cdot k}} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

       12. lower-*.f64 67.4

           \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]

   13. Applied egg-rr 67.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 175000000000:\\ \;\;\;\;\left(\ell \cdot 2\right) \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l k) (/ (* l -0.3333333333333333) (* k t))))

C

double code(double t, double l, double k) {
	return (l / k) * ((l * -0.3333333333333333) / (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 = (l / k) * ((l * (-0.3333333333333333d0)) / (k * t))
end function

Java

public static double code(double t, double l, double k) {
	return (l / k) * ((l * -0.3333333333333333) / (k * t));
}

Python

def code(t, l, k):
	return (l / k) * ((l * -0.3333333333333333) / (k * t))

Julia

function code(t, l, k)
	return Float64(Float64(l / k) * Float64(Float64(l * -0.3333333333333333) / Float64(k * t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / k) * ((l * -0.3333333333333333) / (k * t));
end

Wolfram

code[t_, l_, k_] := N[(N[(l / k), $MachinePrecision] * N[(N[(l * -0.3333333333333333), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-pow.f64 N/A

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

   19. lower-sin.f64 75.7

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

5. Simplified 75.7%

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

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

8. Simplified 46.6%

   \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

9. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

    5. unpow2 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 28.3

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

11. Simplified 28.3%

    \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]
12. Step-by-step derivation

    1. associate-*l* N/A

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

    2. associate-*l* N/A

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

    3. *-commutative N/A

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

    4. lift-*.f64 N/A

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

    5. times-frac N/A

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

    6. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

    7. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{k}} \cdot \frac{\ell \cdot \frac{-1}{3}}{t \cdot k} \]

    8. lower-/.f64 N/A

       \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\ell \cdot \frac{-1}{3}}{t \cdot k}} \]

    9. lower-*.f64 29.6

       \[\leadsto \frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot -0.3333333333333333}}{t \cdot k} \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{t \cdot k}} \]

    11. *-commutative N/A

        \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot \frac{-1}{3}}{\color{blue}{k \cdot t}} \]

    12. lower-*.f64 29.6

        \[\leadsto \frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{\color{blue}{k \cdot t}} \]

13. Applied egg-rr 29.6%

    \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot -0.3333333333333333}{k \cdot t}} \]
14. Add Preprocessing

Alternative 16: 30.8% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (* l (/ l (* t (* k k))))))

C

double code(double t, double l, double k) {
	return -0.3333333333333333 * (l * (l / (t * (k * k))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * (l * (l / (t * (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return -0.3333333333333333 * (l * (l / (t * (k * k))));
}

Python

def code(t, l, k):
	return -0.3333333333333333 * (l * (l / (t * (k * k))))

Julia

function code(t, l, k)
	return Float64(-0.3333333333333333 * Float64(l * Float64(l / Float64(t * Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.3333333333333333 * (l * (l / (t * (k * k))));
end

Wolfram

code[t_, l_, k_] := N[(-0.3333333333333333 * N[(l * N[(l / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\cos k}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   12. associate-*l* N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{\color{blue}{k \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot k\right)}} \]

   15. *-commutative N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}\right)} \]

   18. lower-pow.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{\sin k}^{2}}\right)\right)} \]

   19. lower-sin.f64 75.7

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\color{blue}{\sin k}}^{2}\right)\right)} \]

5. Simplified 75.7%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) \cdot {k}^{2}\right)} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) \cdot {k}^{2} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]

8. Simplified 46.6%

   \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \mathsf{fma}\left(k, k \cdot -0.3333333333333333, 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

9. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot \frac{-1}{3}}}{{k}^{2} \cdot t} \]

    5. unpow2 N/A

       \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\color{blue}{{k}^{2} \cdot t}} \]

    8. unpow2 N/A

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    9. lower-*.f64 28.3

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

11. Simplified 28.3%

    \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t}} \]

12. Taylor expanded in l around 0

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. unpow2 N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

    3. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2} \cdot t}\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2} \cdot t}\right)} \]

    5. lower-/.f64 N/A

       \[\leadsto \frac{-1}{3} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{k}^{2} \cdot t}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{-1}{3} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot t}}\right) \]

    7. unpow2 N/A

       \[\leadsto \frac{-1}{3} \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    8. lower-*.f64 29.0

       \[\leadsto -0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

14. Simplified 29.0%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right)} \]

15. Final simplification 29.0%

    \[\leadsto -0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))