Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 94.0%

Time: 20.0s

Alternatives: 10

Speedup: 28.1×

• 34.8% 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: 35.2% 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: 94.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. associate-*l* 33.3%

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

   2. associate-*l* 33.3%

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

   3. associate-/r* 33.3%

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

   4. associate-/r/ 33.0%

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

   5. *-commutative 33.0%

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

   6. times-frac 33.6%

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

   7. +-commutative 33.6%

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

   8. associate--l+ 42.0%

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

   9. metadata-eval 42.0%

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

   10. +-rgt-identity 42.0%

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

   11. times-frac 45.5%

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

3. Simplified 45.5%

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

4. Taylor expanded in t around 0 77.6%

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

   1. unpow2 77.6%

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

6. Simplified 77.6%

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

   1. associate-*l/ 77.7%

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

   2. associate-*l* 81.5%

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

8. Applied egg-rr 81.5%

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

   1. associate-*l/ 81.5%

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

   2. associate-*r* 86.8%

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

   3. associate-*r/ 86.8%

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

   4. associate-/r* 87.3%

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

   5. associate-/r* 81.9%

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

10. Simplified 81.9%

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

11. Taylor expanded in k around 0 81.5%

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

    1. unpow2 81.5%

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

    2. associate-*r* 86.9%

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

    3. *-commutative 86.9%

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

    4. associate-/r* 94.2%

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

    5. *-commutative 94.2%

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

13. Simplified 94.2%

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

14. Final simplification 94.2%

    \[\leadsto \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k} \cdot \frac{\ell}{\tan k} \]

Alternative 2: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.9e-15)
   (* (/ l k) (/ (* 2.0 (/ (/ l k) (* k t))) (sin k)))
   (* (/ 2.0 (* t (* k k))) (* (/ l (tan k)) (/ l (sin k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.9e-15) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	} else {
		tmp = (2.0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.9d-15) then
        tmp = (l / k) * ((2.0d0 * ((l / k) / (k * t))) / sin(k))
    else
        tmp = (2.0d0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.9e-15) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / Math.sin(k));
	} else {
		tmp = (2.0 / (t * (k * k))) * ((l / Math.tan(k)) * (l / Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.9e-15:
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / math.sin(k))
	else:
		tmp = (2.0 / (t * (k * k))) * ((l / math.tan(k)) * (l / math.sin(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.9e-15)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 * Float64(Float64(l / k) / Float64(k * t))) / sin(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * k))) * Float64(Float64(l / tan(k)) * Float64(l / sin(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.9e-15)
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	else
		tmp = (2.0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.9e-15], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.90000000000000019e-15

   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. Step-by-step derivation

      1. associate-*l* 36.9%

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

      2. associate-*l* 36.9%

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

      3. associate-/r* 36.9%

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

      4. associate-/r/ 36.9%

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

      5. *-commutative 36.9%

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

      6. times-frac 37.7%

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

      7. +-commutative 37.7%

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

      8. associate--l+ 44.3%

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

      9. metadata-eval 44.3%

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

      10. +-rgt-identity 44.3%

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

      11. times-frac 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in t around 0 81.3%

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

      1. unpow2 81.3%

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

   6. Simplified 81.3%

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

      1. associate-*l/ 81.3%

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

      2. associate-*l* 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r* 90.8%

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

      3. associate-*r/ 90.8%

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

      4. associate-/r* 90.9%

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

      5. associate-/r* 84.7%

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

   10. Simplified 84.7%

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

   11. Taylor expanded in k around 0 84.5%

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

       1. unpow2 84.5%

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

       2. associate-*r* 90.9%

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

       3. *-commutative 90.9%

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

       4. associate-/r* 95.5%

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

       5. *-commutative 95.5%

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

   13. Simplified 95.5%

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

   14. Taylor expanded in k around 0 78.4%

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

   if 2.90000000000000019e-15 < k

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. associate-*l* 24.4%

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

      3. associate-/r* 24.4%

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

      4. associate-/r/ 23.5%

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

      5. *-commutative 23.5%

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

      6. times-frac 23.6%

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

      7. +-commutative 23.6%

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

      8. associate--l+ 36.5%

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

      9. metadata-eval 36.5%

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

      10. +-rgt-identity 36.5%

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

      11. times-frac 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around 0 68.7%

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

      1. unpow2 68.7%

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

   6. Simplified 68.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \]

Alternative 3: 74.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.38 \cdot 10^{-58}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.38e-58)
   (* (/ l k) (/ (* 2.0 (/ (/ l k) (* k t))) (sin k)))
   (*
    2.0
    (*
     (/ (cos k) (* k k))
     (* (/ l t) (+ (/ l (* k k)) (* l 0.3333333333333333)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.38e-58) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	}
	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 <= 1.38d-58) then
        tmp = (l / k) * ((2.0d0 * ((l / k) / (k * t))) / sin(k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.38e-58) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / Math.sin(k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.38e-58:
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / math.sin(k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.38e-58)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 * Float64(Float64(l / k) / Float64(k * t))) / sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(Float64(l / Float64(k * k)) + Float64(l * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.38e-58)
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.38e-58], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.37999999999999996e-58

   1. Initial program 37.3%

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

      1. associate-*l* 37.3%

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

      2. associate-*l* 37.3%

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

      3. associate-/r* 37.3%

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

      4. associate-/r/ 37.3%

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

      5. *-commutative 37.3%

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

      6. times-frac 38.1%

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

      7. +-commutative 38.1%

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

      8. associate--l+ 45.1%

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

      9. metadata-eval 45.1%

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

      10. +-rgt-identity 45.1%

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

      11. times-frac 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in t around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 81.9%

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

      2. associate-*l* 86.9%

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

   8. Applied egg-rr 86.9%

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

      1. associate-*l/ 86.9%

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

      2. associate-*r* 90.9%

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

      3. associate-*r/ 90.9%

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

      4. associate-/r* 90.9%

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

      5. associate-/r* 84.4%

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

   10. Simplified 84.4%

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

   11. Taylor expanded in k around 0 84.2%

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

       1. unpow2 84.2%

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

       2. associate-*r* 90.9%

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

       3. *-commutative 90.9%

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

       4. associate-/r* 95.4%

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

       5. *-commutative 95.4%

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

   13. Simplified 95.4%

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

   14. Taylor expanded in k around 0 77.3%

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

   if 1.37999999999999996e-58 < k

   1. Initial program 25.0%

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

      1. +-rgt-identity 22.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]

      2. associate-*l* 22.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]

      3. mul0-rgt 14.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]

      4. distribute-lft-in 23.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]

      5. +-rgt-identity 25.0%

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

      6. sub-neg 25.0%

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

      7. +-commutative 25.0%

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

      8. associate-+l+ 36.4%

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

      9. metadata-eval 36.4%

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

      10. metadata-eval 36.4%

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

      11. +-rgt-identity 36.4%

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

   3. Simplified 36.4%

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

      1. associate-*r* 36.4%

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

      2. associate-*r* 36.4%

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

      3. associate-/l/ 36.4%

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

      4. +-rgt-identity 36.4%

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

      5. div-inv 36.4%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. +-rgt-identity 36.4%

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

      7. *-un-lft-identity 36.4%

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

      8. times-frac 36.4%

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

      9. metadata-eval 36.4%

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

      10. pow-flip 36.5%

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

      11. metadata-eval 36.5%

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

      12. associate-*l/ 36.3%

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

   5. Applied egg-rr 37.8%

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

      1. associate-*r/ 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in k around inf 68.9%

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

      1. times-frac 72.2%

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

      2. unpow2 72.2%

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

      3. unpow2 72.2%

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

      4. *-commutative 72.2%

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

      5. times-frac 79.1%

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

   10. Simplified 79.1%

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

   11. Taylor expanded in k around 0 65.4%

       \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)}\right)\right) \]
   12. Step-by-step derivation

       1. unpow2 65.4%

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

       2. *-commutative 65.4%

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

   13. Simplified 65.4%

       \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.38 \cdot 10^{-58}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]

Alternative 4: 74.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 90:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 90.0)
   (* (/ l k) (/ (* 2.0 (/ (/ l k) (* k t))) (sin k)))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l k) (/ l (* k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 90.0) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / k) * (l / (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 <= 90.0d0) then
        tmp = (l / k) * ((2.0d0 * ((l / k) / (k * t))) / sin(k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / k) * (l / (k * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 90.0) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / Math.sin(k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / k) * (l / (k * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 90.0:
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / math.sin(k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / k) * (l / (k * t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 90.0)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 * Float64(Float64(l / k) / Float64(k * t))) / sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / k) * Float64(l / Float64(k * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 90.0)
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / k) * (l / (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 90.0], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 90

   1. Initial program 37.2%

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

      1. associate-*l* 37.2%

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

      2. associate-*l* 37.2%

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

      3. associate-/r* 37.2%

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

      4. associate-/r/ 37.2%

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

      5. *-commutative 37.2%

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

      6. times-frac 37.9%

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

      7. +-commutative 37.9%

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

      8. associate--l+ 44.4%

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

      9. metadata-eval 44.4%

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

      10. +-rgt-identity 44.4%

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

      11. times-frac 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in t around 0 81.7%

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

      1. unpow2 81.7%

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

   6. Simplified 81.7%

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

      1. associate-*l/ 81.7%

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

      2. associate-*l* 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. associate-*l/ 86.3%

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

      2. associate-*r* 91.0%

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

      3. associate-*r/ 91.0%

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

      4. associate-/r* 91.1%

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

      5. associate-/r* 85.0%

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

   10. Simplified 85.0%

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

   11. Taylor expanded in k around 0 84.9%

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

       1. unpow2 84.9%

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

       2. associate-*r* 91.0%

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

       3. *-commutative 91.0%

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

       4. associate-/r* 95.6%

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

       5. *-commutative 95.6%

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

   13. Simplified 95.6%

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

   14. Taylor expanded in k around 0 78.9%

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

   if 90 < k

   1. Initial program 22.9%

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

      1. +-rgt-identity 22.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]

      2. associate-*l* 22.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]

      3. mul0-rgt 9.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]

      4. distribute-lft-in 21.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]

      5. +-rgt-identity 22.9%

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

      6. sub-neg 22.9%

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

      7. +-commutative 22.9%

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

      8. associate-+l+ 36.5%

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

      9. metadata-eval 36.5%

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

      10. metadata-eval 36.5%

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

      11. +-rgt-identity 36.5%

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

   3. Simplified 36.5%

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

      1. associate-*r* 36.5%

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

      2. associate-*r* 36.5%

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

      3. associate-/l/ 36.5%

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

      4. +-rgt-identity 36.5%

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

      5. div-inv 36.5%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. +-rgt-identity 36.5%

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

      7. *-un-lft-identity 36.5%

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

      8. times-frac 36.5%

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

      9. metadata-eval 36.5%

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

      10. pow-flip 36.6%

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

      11. metadata-eval 36.6%

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

      12. associate-*l/ 36.4%

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

   5. Applied egg-rr 38.1%

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

      1. associate-*r/ 38.2%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in k around inf 66.9%

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

      1. times-frac 70.8%

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

      2. unpow2 70.8%

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

      3. unpow2 70.8%

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

      4. *-commutative 70.8%

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

      5. times-frac 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in k around 0 55.0%

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

       1. unpow2 55.0%

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

       2. unpow2 55.0%

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

       3. associate-*r* 55.0%

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

       4. *-commutative 55.0%

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

       5. times-frac 56.7%

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

       6. *-commutative 56.7%

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

   13. Simplified 56.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 90:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\right)\\ \end{array} \]

Alternative 5: 73.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.9e-15)
   (* (/ l k) (/ (* 2.0 (/ (/ l k) (* k t))) (sin k)))
   (*
    2.0
    (* (* (/ l k) (/ l k)) (/ (+ (/ 1.0 (* k k)) -0.16666666666666666) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.9e-15) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / 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 <= 2.9d-15) then
        tmp = (l / k) * ((2.0d0 * ((l / k) / (k * t))) / sin(k))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (((1.0d0 / (k * k)) + (-0.16666666666666666d0)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.9e-15) {
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / Math.sin(k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.9e-15:
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / math.sin(k))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.9e-15)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 * Float64(Float64(l / k) / Float64(k * t))) / sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.9e-15)
		tmp = (l / k) * ((2.0 * ((l / k) / (k * t))) / sin(k));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.9e-15], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.90000000000000019e-15

   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. Step-by-step derivation

      1. associate-*l* 36.9%

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

      2. associate-*l* 36.9%

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

      3. associate-/r* 36.9%

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

      4. associate-/r/ 36.9%

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

      5. *-commutative 36.9%

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

      6. times-frac 37.7%

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

      7. +-commutative 37.7%

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

      8. associate--l+ 44.3%

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

      9. metadata-eval 44.3%

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

      10. +-rgt-identity 44.3%

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

      11. times-frac 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in t around 0 81.3%

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

      1. unpow2 81.3%

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

   6. Simplified 81.3%

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

      1. associate-*l/ 81.3%

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

      2. associate-*l* 86.1%

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

   8. Applied egg-rr 86.1%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r* 90.8%

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

      3. associate-*r/ 90.8%

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

      4. associate-/r* 90.9%

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

      5. associate-/r* 84.7%

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

   10. Simplified 84.7%

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

   11. Taylor expanded in k around 0 84.5%

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

       1. unpow2 84.5%

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

       2. associate-*r* 90.9%

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

       3. *-commutative 90.9%

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

       4. associate-/r* 95.5%

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

       5. *-commutative 95.5%

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

   13. Simplified 95.5%

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

   14. Taylor expanded in k around 0 78.4%

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

   if 2.90000000000000019e-15 < k

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. associate-*l* 24.4%

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

      3. associate-/r* 24.4%

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

      4. associate-/r/ 23.5%

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

      5. *-commutative 23.5%

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

      6. times-frac 23.6%

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

      7. +-commutative 23.6%

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

      8. associate--l+ 36.5%

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

      9. metadata-eval 36.5%

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

      10. +-rgt-identity 36.5%

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

      11. times-frac 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around 0 68.7%

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

      1. unpow2 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in k around 0 47.9%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 47.9%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 47.9%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 47.9%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 47.9%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 47.9%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

   9. Simplified 47.9%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{k \cdot k}\right)} \]

   10. Taylor expanded in t around 0 47.9%

       \[\leadsto \color{blue}{2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. +-commutative 47.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + -0.16666666666666666 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

       2. *-rgt-identity 47.9%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} + -0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

       3. associate-*r/ 47.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{2}}} + -0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

       4. unpow2 47.9%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \frac{1}{\color{blue}{k \cdot k}} + -0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

       5. *-commutative 47.9%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \frac{1}{k \cdot k} + \color{blue}{{\ell}^{2} \cdot -0.16666666666666666}}{{k}^{2} \cdot t} \]

       6. distribute-lft-in 53.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}}{{k}^{2} \cdot t} \]

       7. unpow2 53.6%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{{k}^{2}}} + -0.16666666666666666\right)}{{k}^{2} \cdot t} \]

       8. metadata-eval 53.6%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} + \color{blue}{\left(-0.16666666666666666\right)}\right)}{{k}^{2} \cdot t} \]

       9. sub-neg 53.6%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}}{{k}^{2} \cdot t} \]

       10. times-frac 53.8%

           \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

       11. unpow2 53.8%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

       12. unpow2 53.8%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

       13. times-frac 56.6%

           \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

   12. Simplified 56.6%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2 \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{t}\right)\\ \end{array} \]

Alternative 6: 71.4% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{t}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (* (* (/ l k) (/ l k)) (/ (+ (/ 1.0 (* k k)) -0.16666666666666666) t))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l / k) * (l / k)) * (((1.0d0 / (k * k)) + (-0.16666666666666666d0)) / t))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t));
}

Python

def code(t, l, k):
	return 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(Float64(1.0 / Float64(k * k)) + -0.16666666666666666) / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) * (l / k)) * (((1.0 / (k * k)) + -0.16666666666666666) / t));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

   1. associate-*l* 33.3%

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

   2. associate-*l* 33.3%

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

   3. associate-/r* 33.3%

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

   4. associate-/r/ 33.0%

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

   5. *-commutative 33.0%

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

   6. times-frac 33.6%

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

   7. +-commutative 33.6%

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

   8. associate--l+ 42.0%

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

   9. metadata-eval 42.0%

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

   10. +-rgt-identity 42.0%

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

   11. times-frac 45.5%

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

3. Simplified 45.5%

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

4. Taylor expanded in t around 0 77.6%

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

   1. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Taylor expanded in k around 0 48.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
8. Step-by-step derivation

   1. *-commutative 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   2. fma-def 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

   3. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   4. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

   5. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

9. Simplified 48.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{k \cdot k}\right)} \]

10. Taylor expanded in t around 0 48.3%

    \[\leadsto \color{blue}{2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}}{{k}^{2} \cdot t}} \]
11. Step-by-step derivation

    1. +-commutative 48.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2}} + -0.16666666666666666 \cdot {\ell}^{2}}}{{k}^{2} \cdot t} \]

    2. *-rgt-identity 48.3%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} + -0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

    3. associate-*r/ 48.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{2}}} + -0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

    4. unpow2 48.3%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \frac{1}{\color{blue}{k \cdot k}} + -0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

    5. *-commutative 48.3%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \frac{1}{k \cdot k} + \color{blue}{{\ell}^{2} \cdot -0.16666666666666666}}{{k}^{2} \cdot t} \]

    6. distribute-lft-in 61.4%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)}}{{k}^{2} \cdot t} \]

    7. unpow2 61.4%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{{k}^{2}}} + -0.16666666666666666\right)}{{k}^{2} \cdot t} \]

    8. metadata-eval 61.4%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} + \color{blue}{\left(-0.16666666666666666\right)}\right)}{{k}^{2} \cdot t} \]

    9. sub-neg 61.4%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}}{{k}^{2} \cdot t} \]

    10. times-frac 61.4%

        \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right)} \]

    11. unpow2 61.4%

        \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

    12. unpow2 61.4%

        \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

    13. times-frac 69.2%

        \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{t}\right) \]

12. Simplified 69.2%

    \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{t}\right)} \]

13. Final simplification 69.2%

    \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{1}{k \cdot k} + -0.16666666666666666}{t}\right) \]

Alternative 7: 69.5% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. associate-*l* 33.3%

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

   2. associate-*l* 33.3%

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

   3. associate-/r* 33.3%

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

   4. associate-/r/ 33.0%

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

   5. *-commutative 33.0%

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

   6. times-frac 33.6%

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

   7. +-commutative 33.6%

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

   8. associate--l+ 42.0%

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

   9. metadata-eval 42.0%

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

   10. +-rgt-identity 42.0%

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

   11. times-frac 45.5%

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

3. Simplified 45.5%

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

4. Taylor expanded in t around 0 77.6%

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

   1. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Taylor expanded in k around 0 61.3%

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

   1. unpow2 61.3%

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

   2. unpow2 61.3%

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

9. Simplified 61.3%

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

10. Taylor expanded in l around 0 61.3%

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

    1. unpow2 61.3%

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

    2. unpow2 61.3%

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

    3. times-frac 68.0%

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

12. Simplified 68.0%

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

13. Final simplification 68.0%

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

Alternative 8: 69.5% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. associate-*l* 33.3%

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

   2. associate-*l* 33.3%

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

   3. associate-/r* 33.3%

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

   4. associate-/r/ 33.0%

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

   5. *-commutative 33.0%

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

   6. times-frac 33.6%

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

   7. +-commutative 33.6%

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

   8. associate--l+ 42.0%

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

   9. metadata-eval 42.0%

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

   10. +-rgt-identity 42.0%

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

   11. times-frac 45.5%

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

3. Simplified 45.5%

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

4. Taylor expanded in t around 0 77.6%

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

   1. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Taylor expanded in k around 0 48.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
8. Step-by-step derivation

   1. *-commutative 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   2. fma-def 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

   3. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   4. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

   5. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

9. Simplified 48.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{k \cdot k}\right)} \]

10. Taylor expanded in l around 0 61.4%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}\right)} \]
11. Step-by-step derivation

    1. *-commutative 61.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)} \]

    2. unpow2 61.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right) \]

    3. associate-*l* 67.8%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)\right)} \]

    4. sub-neg 67.8%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}\right)\right) \]

    5. unpow2 67.8%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)\right)\right) \]

    6. metadata-eval 67.8%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}\right)\right)\right) \]

12. Simplified 67.8%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\right)} \]

13. Taylor expanded in k around 0 61.3%

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

    1. unpow2 61.3%

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

    2. associate-/l* 68.0%

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

    3. unpow2 68.0%

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

15. Simplified 68.0%

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

16. Final simplification 68.0%

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

Alternative 9: 70.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.3%

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

   1. associate-*l* 33.3%

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

   2. associate-*l* 33.3%

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

   3. associate-/r* 33.3%

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

   4. associate-/r/ 33.0%

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

   5. *-commutative 33.0%

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

   6. times-frac 33.6%

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

   7. +-commutative 33.6%

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

   8. associate--l+ 42.0%

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

   9. metadata-eval 42.0%

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

   10. +-rgt-identity 42.0%

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

   11. times-frac 45.5%

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

3. Simplified 45.5%

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

4. Taylor expanded in t around 0 77.6%

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

   1. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Taylor expanded in k around 0 61.3%

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

   1. unpow2 61.3%

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

   2. unpow2 61.3%

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

9. Simplified 61.3%

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

    1. associate-*l/ 61.2%

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

    2. times-frac 68.0%

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

    3. associate-*l* 69.1%

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

11. Applied egg-rr 69.1%

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

12. Final simplification 69.1%

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

Alternative 10: 35.1% accurate, 38.3× speedup

Math

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

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(Float64(Float64(l / t) / 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[(N[(N[(l / t), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

   1. associate-*l* 33.3%

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

   2. associate-*l* 33.3%

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

   3. associate-/r* 33.3%

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

   4. associate-/r/ 33.0%

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

   5. *-commutative 33.0%

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

   6. times-frac 33.6%

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

   7. +-commutative 33.6%

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

   8. associate--l+ 42.0%

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

   9. metadata-eval 42.0%

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

   10. +-rgt-identity 42.0%

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

   11. times-frac 45.5%

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

3. Simplified 45.5%

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

4. Taylor expanded in t around 0 77.6%

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

   1. unpow2 77.6%

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

6. Simplified 77.6%

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

7. Taylor expanded in k around 0 48.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
8. Step-by-step derivation

   1. *-commutative 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   2. fma-def 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

   3. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   4. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

   5. unpow2 48.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

9. Simplified 48.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{k \cdot k}\right)} \]

10. Taylor expanded in k around inf 31.7%

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

    1. unpow2 31.7%

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

    2. associate-*r* 31.9%

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

    3. associate-*r/ 31.9%

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

    4. unpow2 31.9%

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

12. Simplified 31.9%

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

13. Taylor expanded in l around 0 31.7%

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

    1. unpow2 31.7%

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

    2. unpow2 31.7%

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

    3. associate-*r* 31.9%

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

    4. associate-*r/ 32.7%

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

    5. *-commutative 32.7%

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

    6. associate-/r* 33.1%

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

    7. *-commutative 33.1%

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

    8. associate-/r* 32.8%

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

15. Simplified 32.8%

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

16. Final simplification 32.8%

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

Reproduce

herbie shell --seed 2023215 
(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))))