Toniolo and Linder, Equation (10-)

Percentage Accurate: 33.4% → 84.0%

Time: 21.0s

Alternatives: 8

Speedup: 38.3×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 33.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 * (((l / k) / (t / (l / k))) * (cos(k) / (sin(k) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

2. Taylor expanded in t around 0 68.5%

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

   1. associate-*r* 68.5%

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

   2. times-frac 69.2%

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

   3. unpow2 69.2%

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

   4. associate-*l* 70.4%

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

   5. unpow2 70.4%

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

4. Simplified 70.4%

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

   1. associate-*r/ 70.4%

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

   2. times-frac 81.1%

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

6. Applied egg-rr 81.1%

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

   1. div-inv 81.1%

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

8. Applied egg-rr 81.1%

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

   1. associate-*r/ 81.1%

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

   2. *-rgt-identity 81.1%

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

   3. associate-*l/ 83.2%

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

   4. *-commutative 83.2%

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

10. Simplified 83.2%

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

11. Taylor expanded in k around inf 68.7%

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

    1. associate-*r* 68.7%

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

    2. unpow2 68.7%

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

    3. times-frac 69.7%

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

    4. associate-*l* 70.8%

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

    5. *-commutative 70.8%

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

    6. associate-/r* 70.9%

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

    7. unpow2 70.9%

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

    8. associate-*l/ 77.9%

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

    9. *-commutative 77.9%

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

    10. associate-/l/ 77.4%

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

    11. associate-*r/ 79.2%

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

    12. associate-/l* 80.7%

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

    13. associate-/l* 81.6%

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

    14. *-commutative 81.6%

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

    15. associate-/l* 83.7%

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

13. Simplified 83.7%

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

14. Final simplification 83.7%

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

Alternative 2: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\cos k}}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6e-16)
   (/ 2.0 (/ (* (* (/ k l) (* t (/ k l))) (* k k)) (cos k)))
   (/
    2.0
    (/ (* (* k (* k t)) (/ (- 0.5 (/ (cos (+ k k)) 2.0)) (cos k))) (* l l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-16) {
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / cos(k));
	} else {
		tmp = 2.0 / (((k * (k * t)) * ((0.5 - (cos((k + k)) / 2.0)) / cos(k))) / (l * l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6d-16) then
        tmp = 2.0d0 / ((((k / l) * (t * (k / l))) * (k * k)) / cos(k))
    else
        tmp = 2.0d0 / (((k * (k * t)) * ((0.5d0 - (cos((k + k)) / 2.0d0)) / cos(k))) / (l * l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6e-16) {
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / Math.cos(k));
	} else {
		tmp = 2.0 / (((k * (k * t)) * ((0.5 - (Math.cos((k + k)) / 2.0)) / Math.cos(k))) / (l * l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6e-16:
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / math.cos(k))
	else:
		tmp = 2.0 / (((k * (k * t)) * ((0.5 - (math.cos((k + k)) / 2.0)) / math.cos(k))) / (l * l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6e-16)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(t * Float64(k / l))) * Float64(k * k)) / cos(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t)) * Float64(Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0)) / cos(k))) / Float64(l * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6e-16)
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / cos(k));
	else
		tmp = 2.0 / (((k * (k * t)) * ((0.5 - (cos((k + k)) / 2.0)) / cos(k))) / (l * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6e-16], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999987e-16

   1. Initial program 36.9%

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

   2. Taylor expanded in t around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. times-frac 74.0%

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

      3. unpow2 74.0%

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

      4. associate-*l* 75.1%

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

      5. unpow2 75.1%

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

   4. Simplified 75.1%

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

      1. associate-*r/ 75.1%

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

      2. times-frac 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. div-inv 87.2%

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

   8. Applied egg-rr 87.2%

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

      1. associate-*r/ 87.2%

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

      2. *-rgt-identity 87.2%

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

      3. associate-*l/ 89.2%

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

      4. *-commutative 89.2%

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

   10. Simplified 89.2%

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

   11. Taylor expanded in k around 0 82.5%

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

       1. unpow2 82.5%

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

   13. Simplified 82.5%

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

   if 5.99999999999999987e-16 < k

   1. Initial program 21.1%

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

   2. Taylor expanded in t around 0 58.2%

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

      1. associate-*r* 58.2%

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

      2. times-frac 58.2%

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

      3. unpow2 58.2%

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

      4. associate-*l* 59.4%

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

      5. unpow2 59.4%

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

   4. Simplified 59.4%

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

      1. associate-*l/ 59.4%

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

   6. Applied egg-rr 59.4%

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

      1. unpow2 59.4%

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

      2. sin-mult 59.4%

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

   8. Applied egg-rr 59.4%

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

      1. div-sub 59.4%

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

      2. +-inverses 59.4%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}}{\cos k}}{\ell \cdot \ell}} \]

      3. cos-0 59.4%

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

      4. metadata-eval 59.4%

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

   10. Simplified 59.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\cos k}}{\ell \cdot \ell}}\\ \end{array} \]

Alternative 3: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;k \leq 2500:\\ \;\;\;\;\frac{2}{\frac{t_1 \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1 \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ k l) (* t (/ k l)))))
   (if (<= k 2500.0)
     (/ 2.0 (/ (* t_1 (* k k)) (cos k)))
     (/ 2.0 (/ (* t_1 (- 0.5 (/ (cos (+ k k)) 2.0))) (cos k))))))

C

double code(double t, double l, double k) {
	double t_1 = (k / l) * (t * (k / l));
	double tmp;
	if (k <= 2500.0) {
		tmp = 2.0 / ((t_1 * (k * k)) / cos(k));
	} else {
		tmp = 2.0 / ((t_1 * (0.5 - (cos((k + k)) / 2.0))) / cos(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / l) * (t * (k / l))
    if (k <= 2500.0d0) then
        tmp = 2.0d0 / ((t_1 * (k * k)) / cos(k))
    else
        tmp = 2.0d0 / ((t_1 * (0.5d0 - (cos((k + k)) / 2.0d0))) / cos(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k / l) * (t * (k / l));
	double tmp;
	if (k <= 2500.0) {
		tmp = 2.0 / ((t_1 * (k * k)) / Math.cos(k));
	} else {
		tmp = 2.0 / ((t_1 * (0.5 - (Math.cos((k + k)) / 2.0))) / Math.cos(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k / l) * (t * (k / l))
	tmp = 0
	if k <= 2500.0:
		tmp = 2.0 / ((t_1 * (k * k)) / math.cos(k))
	else:
		tmp = 2.0 / ((t_1 * (0.5 - (math.cos((k + k)) / 2.0))) / math.cos(k))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k / l) * Float64(t * Float64(k / l)))
	tmp = 0.0
	if (k <= 2500.0)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(k * k)) / cos(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0))) / cos(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / l) * (t * (k / l));
	tmp = 0.0;
	if (k <= 2500.0)
		tmp = 2.0 / ((t_1 * (k * k)) / cos(k));
	else
		tmp = 2.0 / ((t_1 * (0.5 - (cos((k + k)) / 2.0))) / cos(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2500.0], N[(2.0 / N[(N[(t$95$1 * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2500

   1. Initial program 36.7%

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

   2. Taylor expanded in t around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. times-frac 74.1%

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

      3. unpow2 74.1%

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

      4. associate-*l* 75.2%

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

      5. unpow2 75.2%

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

   4. Simplified 75.2%

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

      1. associate-*r/ 75.2%

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

      2. times-frac 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. div-inv 87.3%

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

   8. Applied egg-rr 87.3%

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

      1. associate-*r/ 87.3%

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

      2. *-rgt-identity 87.3%

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

      3. associate-*l/ 89.3%

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

      4. *-commutative 89.3%

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

   10. Simplified 89.3%

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

   11. Taylor expanded in k around 0 82.6%

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

       1. unpow2 82.6%

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

   13. Simplified 82.6%

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

   if 2500 < k

   1. Initial program 21.4%

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

   2. Taylor expanded in t around 0 57.6%

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

      1. associate-*r* 57.6%

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

      2. times-frac 57.7%

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

      3. unpow2 57.7%

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

      4. associate-*l* 58.9%

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

      5. unpow2 58.9%

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

   4. Simplified 58.9%

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

      1. associate-*r/ 58.9%

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

      2. times-frac 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. div-inv 66.5%

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

   8. Applied egg-rr 66.5%

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

      1. associate-*r/ 66.6%

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

      2. *-rgt-identity 66.6%

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

      3. associate-*l/ 68.9%

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

      4. *-commutative 68.9%

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

   10. Simplified 68.9%

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

       1. unpow2 58.9%

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

       2. sin-mult 58.8%

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

   12. Applied egg-rr 68.9%

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

       1. div-sub 58.8%

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

       2. +-inverses 58.8%

          \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}}{\cos k}}{\ell \cdot \ell}} \]

       3. cos-0 58.8%

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

       4. metadata-eval 58.8%

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

   14. Simplified 68.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2500:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 4: 71.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

2. Taylor expanded in t around 0 68.5%

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

   1. associate-*r* 68.5%

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

   2. times-frac 69.2%

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

   3. unpow2 69.2%

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

   4. associate-*l* 70.4%

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

   5. unpow2 70.4%

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

4. Simplified 70.4%

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

   1. associate-*r/ 70.4%

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

   2. times-frac 81.1%

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

6. Applied egg-rr 81.1%

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

   1. div-inv 81.1%

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

8. Applied egg-rr 81.1%

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

   1. associate-*r/ 81.1%

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

   2. *-rgt-identity 81.1%

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

   3. associate-*l/ 83.2%

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

   4. *-commutative 83.2%

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

10. Simplified 83.2%

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

11. Taylor expanded in k around 0 72.1%

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

    1. unpow2 72.1%

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

13. Simplified 72.1%

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

14. Final simplification 72.1%

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

Alternative 5: 68.0% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ l (/ (pow k 3.0) l)) (* 2.0 (/ 1.0 (* k t)))))

C

double code(double t, double l, double k) {
	return (l / (pow(k, 3.0) / l)) * (2.0 * (1.0 / (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 ** 3.0d0) / l)) * (2.0d0 * (1.0d0 / (k * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 36.8%

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

   4. associate-*r/ 37.6%

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

   5. associate-/l* 36.8%

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

   6. +-commutative 36.8%

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

   7. unpow2 36.8%

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

   8. sqr-neg 36.8%

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

   9. distribute-frac-neg 36.8%

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

   10. distribute-frac-neg 36.8%

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

   11. unpow2 36.8%

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

   12. associate--l+ 49.5%

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

   13. metadata-eval 49.5%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 49.5%

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

   15. unpow2 49.5%

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

   16. distribute-frac-neg 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in k around 0 48.1%

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

5. Taylor expanded in k around inf 62.3%

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

   1. *-commutative 62.3%

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

   2. times-frac 62.8%

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

   3. associate-*l* 62.8%

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

   4. unpow2 62.8%

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

   5. associate-/r* 62.8%

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

7. Simplified 62.8%

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

8. Taylor expanded in k around 0 62.7%

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

9. Taylor expanded in l around 0 62.7%

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

    1. unpow2 62.7%

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

    2. associate-/l* 69.4%

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

11. Simplified 69.4%

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

12. Final simplification 69.4%

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

Alternative 6: 65.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

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 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

2. Taylor expanded in t around 0 68.5%

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

   1. associate-*r* 68.5%

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

   2. times-frac 69.2%

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

   3. unpow2 69.2%

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

   4. associate-*l* 70.4%

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

   5. unpow2 70.4%

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

4. Simplified 70.4%

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

5. Taylor expanded in k around 0 61.8%

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

   1. unpow2 61.8%

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

   2. *-commutative 61.8%

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

   3. times-frac 66.6%

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

7. Simplified 66.6%

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

8. Final simplification 66.6%

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

Alternative 7: 33.2% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (* (/ l t) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l / t) * (l / (k * k)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * ((l / t) * (l / (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l / t) * (l / (k * k)));
}

Python

def code(t, l, k):
	return -0.3333333333333333 * ((l / t) * (l / (k * k)))

Julia

function code(t, l, k)
	return Float64(-0.3333333333333333 * Float64(Float64(l / t) * Float64(l / Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.3333333333333333 * ((l / t) * (l / (k * k)));
end

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 36.8%

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

   4. associate-*r/ 37.6%

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

   5. associate-/l* 36.8%

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

   6. +-commutative 36.8%

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

   7. unpow2 36.8%

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

   8. sqr-neg 36.8%

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

   9. distribute-frac-neg 36.8%

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

   10. distribute-frac-neg 36.8%

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

   11. unpow2 36.8%

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

   12. associate--l+ 49.5%

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

   13. metadata-eval 49.5%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 49.5%

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

   15. unpow2 49.5%

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

   16. distribute-frac-neg 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in k around 0 35.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. fma-def 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. unpow2 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   5. *-commutative 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

6. Simplified 35.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

7. Taylor expanded in k around inf 31.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 31.0%

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

   2. associate-*r* 31.1%

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

   3. associate-*r/ 31.1%

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

   4. *-commutative 31.1%

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

   5. times-frac 31.2%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k \cdot t} \cdot \frac{{\ell}^{2}}{k}} \]

   6. unpow2 31.2%

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

9. Simplified 31.2%

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

10. Taylor expanded in k around 0 31.0%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 31.0%

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

    2. associate-*r* 31.1%

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

    3. *-commutative 31.1%

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

    4. associate-/r* 31.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{k \cdot t}}{k}} \]

    5. unpow2 31.2%

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

12. Simplified 31.2%

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

13. Taylor expanded in l around 0 31.0%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. unpow2 31.0%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

    2. times-frac 31.8%

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

    3. unpow2 31.8%

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

15. Simplified 31.8%

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

16. Final simplification 31.8%

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

Alternative 8: 34.1% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{k}}} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (/ l k) (/ t (/ l k))) -0.3333333333333333))

C

double code(double t, double l, double k) {
	return ((l / k) / (t / (l / k))) * -0.3333333333333333;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / k) / (t / (l / k))) * (-0.3333333333333333d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((l / k) / (t / (l / k))) * -0.3333333333333333;
}

Python

def code(t, l, k):
	return ((l / k) / (t / (l / k))) * -0.3333333333333333

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / k) / Float64(t / Float64(l / k))) * -0.3333333333333333)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / k) / (t / (l / k))) * -0.3333333333333333;
end

Wolfram

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

Derivation

1. Initial program 32.2%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 36.8%

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

   4. associate-*r/ 37.6%

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

   5. associate-/l* 36.8%

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

   6. +-commutative 36.8%

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

   7. unpow2 36.8%

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

   8. sqr-neg 36.8%

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

   9. distribute-frac-neg 36.8%

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

   10. distribute-frac-neg 36.8%

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

   11. unpow2 36.8%

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

   12. associate--l+ 49.5%

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

   13. metadata-eval 49.5%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 49.5%

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

   15. unpow2 49.5%

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

   16. distribute-frac-neg 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in k around 0 35.4%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. fma-def 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. unpow2 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   5. *-commutative 35.4%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}, 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

6. Simplified 35.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}, 2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\right)} \]

7. Taylor expanded in k around inf 31.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 31.0%

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

   2. associate-*r* 31.1%

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

   3. associate-*r/ 31.1%

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

   4. *-commutative 31.1%

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

   5. times-frac 31.2%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k \cdot t} \cdot \frac{{\ell}^{2}}{k}} \]

   6. unpow2 31.2%

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

9. Simplified 31.2%

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

10. Taylor expanded in k around 0 31.0%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 31.0%

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

    2. associate-*r* 31.1%

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

    3. *-commutative 31.1%

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

    4. associate-/r* 31.2%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{k \cdot t}}{k}} \]

    5. unpow2 31.2%

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

12. Simplified 31.2%

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

13. Taylor expanded in l around 0 31.0%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. *-commutative 31.0%

       \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

    2. associate-/r* 31.0%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}} \]

    3. unpow2 31.0%

       \[\leadsto -0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{\color{blue}{k \cdot k}} \]

    4. associate-/l/ 31.3%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\frac{{\ell}^{2}}{t}}{k}}{k}} \]

    5. unpow2 31.3%

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

    6. associate-*r/ 31.9%

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

    7. associate-*l/ 32.3%

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

    8. associate-/l* 32.3%

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

    9. associate-/l* 32.1%

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

    10. *-commutative 32.1%

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

    11. associate-/l* 32.4%

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

15. Simplified 32.4%

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

16. Final simplification 32.4%

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

Reproduce

herbie shell --seed 2023292 
(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))))