Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.7% → 95.4%

Time: 21.2s

Alternatives: 14

Speedup: 28.1×

• 34.7% 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: 34.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

   1. clear-num 80.9%

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

   2. frac-times 80.9%

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

   3. *-un-lft-identity 80.9%

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

8. Applied egg-rr 80.9%

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

   1. frac-times 85.4%

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

   2. associate-*l* 91.1%

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

   3. *-commutative 91.1%

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

10. Applied egg-rr 91.1%

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

    1. *-commutative 91.1%

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

    2. associate-*l* 92.9%

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

    3. times-frac 96.0%

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

    4. associate-*r* 95.3%

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

12. Simplified 95.3%

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

    1. associate-*r/ 95.2%

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

    2. associate-*l* 96.2%

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

14. Applied egg-rr 96.2%

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

15. Final simplification 96.2%

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

Alternative 2: 94.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \tan k\\ \mathbf{if}\;k \leq 4 \cdot 10^{+211}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\ell \cdot \frac{\ell}{k}}{\sin k \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (tan k))))
   (if (<= k 4e+211)
     (* (/ 2.0 k) (* (/ (/ l (sin k)) k) (/ l t_1)))
     (* (/ 2.0 k) (/ (* l (/ l k)) (* (sin k) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = t * tan(k);
	double tmp;
	if (k <= 4e+211) {
		tmp = (2.0 / k) * (((l / sin(k)) / k) * (l / t_1));
	} else {
		tmp = (2.0 / k) * ((l * (l / k)) / (sin(k) * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * tan(k)
    if (k <= 4d+211) then
        tmp = (2.0d0 / k) * (((l / sin(k)) / k) * (l / t_1))
    else
        tmp = (2.0d0 / k) * ((l * (l / k)) / (sin(k) * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.tan(k);
	double tmp;
	if (k <= 4e+211) {
		tmp = (2.0 / k) * (((l / Math.sin(k)) / k) * (l / t_1));
	} else {
		tmp = (2.0 / k) * ((l * (l / k)) / (Math.sin(k) * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = t * math.tan(k)
	tmp = 0
	if k <= 4e+211:
		tmp = (2.0 / k) * (((l / math.sin(k)) / k) * (l / t_1))
	else:
		tmp = (2.0 / k) * ((l * (l / k)) / (math.sin(k) * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(t * tan(k))
	tmp = 0.0
	if (k <= 4e+211)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / sin(k)) / k) * Float64(l / t_1)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l * Float64(l / k)) / Float64(sin(k) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = t * tan(k);
	tmp = 0.0;
	if (k <= 4e+211)
		tmp = (2.0 / k) * (((l / sin(k)) / k) * (l / t_1));
	else
		tmp = (2.0 / k) * ((l * (l / k)) / (sin(k) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4e+211], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.9999999999999998e211

   1. Initial program 33.4%

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

      1. associate-*l* 33.4%

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

      2. associate-*l* 33.4%

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

      3. associate-/r* 33.4%

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

      4. associate-/r/ 33.4%

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

      5. *-commutative 33.4%

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

      6. times-frac 33.8%

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

      7. +-commutative 33.8%

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

      8. associate--l+ 40.5%

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

      9. metadata-eval 40.5%

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

      10. +-rgt-identity 40.5%

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

      11. times-frac 44.8%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in t around 0 81.8%

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

      1. unpow2 81.8%

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

   6. Simplified 81.8%

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

      1. associate-*l/ 81.8%

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

      2. associate-*l* 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. times-frac 87.5%

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

   10. Simplified 87.5%

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

       1. *-un-lft-identity 87.5%

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

       2. times-frac 93.0%

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

   12. Applied egg-rr 93.0%

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

       1. *-lft-identity 93.0%

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

       2. associate-/l/ 92.2%

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

   14. Simplified 92.2%

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

   if 3.9999999999999998e211 < k

   1. Initial program 52.6%

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

      1. associate-*l* 52.6%

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

      2. associate-*l* 52.6%

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

      3. associate-/r* 52.6%

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

      4. associate-/r/ 52.6%

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

      5. *-commutative 52.6%

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

      6. times-frac 52.6%

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

      7. +-commutative 52.6%

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

      8. associate--l+ 57.9%

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

      9. metadata-eval 57.9%

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

      10. +-rgt-identity 57.9%

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

      11. times-frac 57.9%

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

   3. Simplified 57.9%

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

   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. Step-by-step derivation

      1. associate-*l/ 68.7%

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

      2. associate-*l* 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. times-frac 84.0%

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

   10. Simplified 84.0%

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

       1. *-un-lft-identity 84.0%

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

       2. times-frac 89.7%

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

   12. Applied egg-rr 89.7%

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

       1. *-lft-identity 89.7%

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

       2. associate-/l/ 89.7%

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

       3. associate-/r* 89.7%

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

       4. associate-/l/ 89.7%

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

   14. Simplified 89.7%

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

       1. frac-times 99.8%

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

   16. Applied egg-rr 99.8%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+211}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \frac{\ell}{t \cdot \tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\ell \cdot \frac{\ell}{k}}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \end{array} \]

Alternative 3: 94.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

   1. associate-*l/ 80.9%

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

   2. associate-*l* 86.5%

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

8. Applied egg-rr 86.5%

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

   1. times-frac 87.2%

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

10. Simplified 87.2%

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

    1. *-un-lft-identity 87.2%

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

    2. times-frac 92.8%

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

12. Applied egg-rr 92.8%

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

    1. *-lft-identity 92.8%

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

    2. associate-/l/ 90.6%

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

    3. associate-/r* 92.8%

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

    4. associate-/l/ 92.0%

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

14. Simplified 92.0%

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

15. Final simplification 92.0%

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

Alternative 4: 94.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

   1. associate-*l/ 80.9%

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

   2. associate-*l* 86.5%

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

8. Applied egg-rr 86.5%

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

   1. times-frac 87.2%

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

10. Simplified 87.2%

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

    1. *-un-lft-identity 87.2%

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

    2. times-frac 92.8%

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

12. Applied egg-rr 92.8%

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

    1. *-lft-identity 92.8%

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

    2. associate-/l/ 92.1%

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

14. Simplified 92.1%

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

15. Final simplification 92.1%

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

Alternative 5: 94.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

   1. clear-num 80.9%

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

   2. frac-times 80.9%

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

   3. *-un-lft-identity 80.9%

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

8. Applied egg-rr 80.9%

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

   1. frac-times 85.4%

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

   2. associate-*l* 91.1%

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

   3. *-commutative 91.1%

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

10. Applied egg-rr 91.1%

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

    1. *-commutative 91.1%

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

    2. associate-*l* 92.9%

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

    3. times-frac 96.0%

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

    4. associate-*r* 95.3%

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

12. Simplified 95.3%

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

13. Final simplification 95.3%

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

Alternative 6: 73.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-125)
   (/ (* (/ 2.0 k) (pow (/ l k) 2.0)) (* k t))
   (*
    2.0
    (/
     (* (cos k) (* (/ l t) (+ (/ l (* k k)) (* l 0.3333333333333333))))
     (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-125) {
		tmp = ((2.0 / k) * pow((l / k), 2.0)) / (k * t);
	} else {
		tmp = 2.0 * ((cos(k) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333)))) / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.5d-125) then
        tmp = ((2.0d0 / k) * ((l / k) ** 2.0d0)) / (k * t)
    else
        tmp = 2.0d0 * ((cos(k) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333d0)))) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-125) {
		tmp = ((2.0 / k) * Math.pow((l / k), 2.0)) / (k * t);
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333)))) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.5e-125:
		tmp = ((2.0 / k) * math.pow((l / k), 2.0)) / (k * t)
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333)))) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-125)
		tmp = Float64(Float64(Float64(2.0 / k) * (Float64(l / k) ^ 2.0)) / Float64(k * t));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / t) * Float64(Float64(l / Float64(k * k)) + Float64(l * 0.3333333333333333)))) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.5e-125)
		tmp = ((2.0 / k) * ((l / k) ^ 2.0)) / (k * t);
	else
		tmp = 2.0 * ((cos(k) * ((l / t) * ((l / (k * k)) + (l * 0.3333333333333333)))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.5e-125], N[(N[(N[(2.0 / k), $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.5000000000000002e-125

   1. Initial program 32.2%

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

      1. associate-*l* 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-/r* 32.2%

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

      4. associate-/r/ 32.3%

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

      5. *-commutative 32.3%

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

      6. times-frac 32.8%

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

      7. +-commutative 32.8%

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

      8. associate--l+ 38.7%

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

      9. metadata-eval 38.7%

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

      10. +-rgt-identity 38.7%

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

      11. times-frac 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in t around 0 83.5%

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

      1. unpow2 83.5%

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

   6. Simplified 83.5%

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

      1. associate-*l/ 83.5%

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

      2. associate-*l* 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. times-frac 90.1%

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

   10. Simplified 90.1%

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

   11. Taylor expanded in k around 0 63.4%

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

       1. unpow2 63.4%

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

       2. unpow2 63.4%

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

   13. Simplified 63.4%

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

       1. associate-*r/ 63.4%

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

       2. times-frac 81.5%

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

       3. pow1 81.5%

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

       4. pow1 81.5%

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

       5. pow-prod-up 81.5%

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

       6. metadata-eval 81.5%

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

   15. Applied egg-rr 81.5%

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

   if 8.5000000000000002e-125 < k

   1. Initial program 38.6%

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

      1. associate-*l* 38.6%

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

      2. associate-*l* 38.6%

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

      3. associate-/r* 38.6%

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

      4. associate-/r/ 38.6%

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

      5. *-commutative 38.6%

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

      6. times-frac 38.7%

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

      7. +-commutative 38.7%

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

      8. associate--l+ 46.4%

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

      9. metadata-eval 46.4%

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

      10. +-rgt-identity 46.4%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in t around 0 75.9%

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

      1. times-frac 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

      4. *-commutative 76.9%

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

   6. Simplified 76.9%

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

      1. associate-*l/ 76.9%

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

      2. times-frac 84.4%

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

   8. Applied egg-rr 84.4%

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

   9. Taylor expanded in k around 0 74.6%

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

       1. unpow2 74.6%

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

       2. *-commutative 74.6%

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

   11. Simplified 74.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)}{k \cdot k}\\ \end{array} \]

Alternative 7: 73.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.5e-125)
   (/ (* (/ 2.0 k) (pow (/ l k) 2.0)) (* k t))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l t) (/ l (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-125) {
		tmp = ((2.0 / k) * pow((l / k), 2.0)) / (k * t);
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.5d-125) then
        tmp = ((2.0d0 / k) * ((l / k) ** 2.0d0)) / (k * t)
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.5e-125) {
		tmp = ((2.0 / k) * Math.pow((l / k), 2.0)) / (k * t);
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.5e-125:
		tmp = ((2.0 / k) * math.pow((l / k), 2.0)) / (k * t)
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.5e-125)
		tmp = Float64(Float64(Float64(2.0 / k) * (Float64(l / k) ^ 2.0)) / Float64(k * t));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.5e-125)
		tmp = ((2.0 / k) * ((l / k) ^ 2.0)) / (k * t);
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.5e-125], N[(N[(N[(2.0 / k), $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.5000000000000002e-125

   1. Initial program 32.2%

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

      1. associate-*l* 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-/r* 32.2%

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

      4. associate-/r/ 32.3%

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

      5. *-commutative 32.3%

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

      6. times-frac 32.8%

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

      7. +-commutative 32.8%

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

      8. associate--l+ 38.7%

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

      9. metadata-eval 38.7%

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

      10. +-rgt-identity 38.7%

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

      11. times-frac 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in t around 0 83.5%

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

      1. unpow2 83.5%

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

   6. Simplified 83.5%

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

      1. associate-*l/ 83.5%

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

      2. associate-*l* 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. times-frac 90.1%

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

   10. Simplified 90.1%

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

   11. Taylor expanded in k around 0 63.4%

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

       1. unpow2 63.4%

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

       2. unpow2 63.4%

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

   13. Simplified 63.4%

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

       1. associate-*r/ 63.4%

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

       2. times-frac 81.5%

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

       3. pow1 81.5%

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

       4. pow1 81.5%

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

       5. pow-prod-up 81.5%

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

       6. metadata-eval 81.5%

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

   15. Applied egg-rr 81.5%

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

   if 8.5000000000000002e-125 < k

   1. Initial program 38.6%

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

      1. associate-*l* 38.6%

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

      2. associate-*l* 38.6%

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

      3. associate-/r* 38.6%

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

      4. associate-/r/ 38.6%

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

      5. *-commutative 38.6%

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

      6. times-frac 38.7%

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

      7. +-commutative 38.7%

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

      8. associate--l+ 46.4%

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

      9. metadata-eval 46.4%

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

      10. +-rgt-identity 46.4%

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

      11. times-frac 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in t around 0 75.9%

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

      1. times-frac 76.9%

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

      2. unpow2 76.9%

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

      3. unpow2 76.9%

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

      4. *-commutative 76.9%

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

   6. Simplified 76.9%

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

   7. Taylor expanded in k around 0 66.4%

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

      1. unpow2 66.4%

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

      2. *-commutative 66.4%

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

      3. times-frac 72.5%

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

      4. unpow2 72.5%

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

   9. Simplified 72.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 8: 74.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.45e+16)
   (* (/ 2.0 k) (* (/ (/ l k) (sin k)) (/ (/ l k) t)))
   (* -0.3333333333333333 (/ (pow (/ l k) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.45e+16) {
		tmp = (2.0 / k) * (((l / k) / sin(k)) * ((l / k) / t));
	} else {
		tmp = -0.3333333333333333 * (pow((l / k), 2.0) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.45d+16) then
        tmp = (2.0d0 / k) * (((l / k) / sin(k)) * ((l / k) / t))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) ** 2.0d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.45e+16) {
		tmp = (2.0 / k) * (((l / k) / Math.sin(k)) * ((l / k) / t));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / k), 2.0) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.45e+16:
		tmp = (2.0 / k) * (((l / k) / math.sin(k)) * ((l / k) / t))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / k), 2.0) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.45e+16)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / k) / sin(k)) * Float64(Float64(l / k) / t)));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / k) ^ 2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.45e+16)
		tmp = (2.0 / k) * (((l / k) / sin(k)) * ((l / k) / t));
	else
		tmp = -0.3333333333333333 * (((l / k) ^ 2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.45e+16], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e16

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-/r* 32.4%

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

      4. associate-/r/ 32.4%

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

      5. *-commutative 32.4%

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

      6. times-frac 33.0%

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

      7. +-commutative 33.0%

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

      8. associate--l+ 40.1%

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

      9. metadata-eval 40.1%

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

      10. +-rgt-identity 40.1%

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

      11. times-frac 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 82.6%

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

      1. unpow2 82.6%

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

   6. Simplified 82.6%

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

      1. associate-*l/ 82.6%

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

      2. associate-*l* 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. times-frac 88.2%

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

   10. Simplified 88.2%

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

       1. *-un-lft-identity 88.2%

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

       2. times-frac 92.9%

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

   12. Applied egg-rr 92.9%

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

       1. *-lft-identity 92.9%

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

       2. associate-/l/ 90.0%

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

       3. associate-/r* 92.9%

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

       4. associate-/l/ 92.0%

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

   14. Simplified 92.0%

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

   15. Taylor expanded in k around 0 80.5%

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

       1. associate-/r* 81.5%

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

   17. Simplified 81.5%

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

   if 2.45e16 < k

   1. Initial program 42.0%

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

      1. associate-*l* 42.0%

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

      2. associate-*l* 42.0%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 42.0%

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

      7. +-commutative 42.0%

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

      8. associate--l+ 47.0%

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

      9. metadata-eval 47.0%

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

      10. +-rgt-identity 47.0%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 75.6%

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

      1. unpow2 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in k around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. fma-def 54.6%

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

      3. unpow2 54.6%

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

      4. unpow2 54.6%

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

      5. unpow2 54.6%

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

   9. Simplified 54.6%

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

   10. Taylor expanded in k around inf 59.4%

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

       1. associate-/r* 59.4%

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

       2. unpow2 59.4%

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

       3. unpow2 59.4%

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

       4. times-frac 60.9%

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

       5. unpow2 60.9%

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

   12. Simplified 60.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 9: 73.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2}{\frac{\sin k}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e+16)
   (* (/ l k) (/ 2.0 (* (/ (sin k) l) (* k (* k t)))))
   (* -0.3333333333333333 (/ (pow (/ l k) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e+16) {
		tmp = (l / k) * (2.0 / ((sin(k) / l) * (k * (k * t))));
	} else {
		tmp = -0.3333333333333333 * (pow((l / k), 2.0) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d+16) then
        tmp = (l / k) * (2.0d0 / ((sin(k) / l) * (k * (k * t))))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) ** 2.0d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e+16) {
		tmp = (l / k) * (2.0 / ((Math.sin(k) / l) * (k * (k * t))));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / k), 2.0) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.2e+16:
		tmp = (l / k) * (2.0 / ((math.sin(k) / l) * (k * (k * t))))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / k), 2.0) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e+16)
		tmp = Float64(Float64(l / k) * Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(k * Float64(k * t)))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / k) ^ 2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e+16)
		tmp = (l / k) * (2.0 / ((sin(k) / l) * (k * (k * t))));
	else
		tmp = -0.3333333333333333 * (((l / k) ^ 2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.2e+16], N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e16

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-/r* 32.4%

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

      4. associate-/r/ 32.4%

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

      5. *-commutative 32.4%

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

      6. times-frac 33.0%

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

      7. +-commutative 33.0%

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

      8. associate--l+ 40.1%

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

      9. metadata-eval 40.1%

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

      10. +-rgt-identity 40.1%

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

      11. times-frac 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 82.6%

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

      1. unpow2 82.6%

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

   6. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. frac-times 82.6%

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

      3. *-un-lft-identity 82.6%

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

   8. Applied egg-rr 82.6%

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

      1. frac-times 87.4%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   10. Applied egg-rr 93.5%

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

       1. *-commutative 93.5%

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

       2. associate-*l* 94.4%

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

       3. times-frac 97.2%

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

       4. associate-*r* 96.3%

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

   12. Simplified 96.3%

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

   13. Taylor expanded in k around 0 78.6%

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

       1. unpow2 78.6%

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

       2. associate-*r* 81.6%

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

   15. Simplified 81.6%

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

   if 2.2e16 < k

   1. Initial program 42.0%

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

      1. associate-*l* 42.0%

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

      2. associate-*l* 42.0%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 42.0%

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

      7. +-commutative 42.0%

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

      8. associate--l+ 47.0%

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

      9. metadata-eval 47.0%

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

      10. +-rgt-identity 47.0%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 75.6%

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

      1. unpow2 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in k around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. fma-def 54.6%

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

      3. unpow2 54.6%

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

      4. unpow2 54.6%

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

      5. unpow2 54.6%

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

   9. Simplified 54.6%

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

   10. Taylor expanded in k around inf 59.4%

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

       1. associate-/r* 59.4%

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

       2. unpow2 59.4%

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

       3. unpow2 59.4%

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

       4. times-frac 60.9%

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

       5. unpow2 60.9%

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

   12. Simplified 60.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{2}{\frac{\sin k}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 10: 72.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

   1. clear-num 80.9%

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

   2. frac-times 80.9%

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

   3. *-un-lft-identity 80.9%

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

8. Applied egg-rr 80.9%

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

   1. frac-times 85.4%

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

   2. associate-*l* 91.1%

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

   3. *-commutative 91.1%

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

10. Applied egg-rr 91.1%

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

    1. *-commutative 91.1%

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

    2. associate-*l* 92.9%

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

    3. times-frac 96.0%

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

    4. associate-*r* 95.3%

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

12. Simplified 95.3%

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

13. Taylor expanded in k around 0 76.8%

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

14. Final simplification 76.8%

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

Alternative 11: 72.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;k \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot t_1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{t_1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ l k) 2.0)))
   (if (<= k 2.45e+16)
     (/ (* (/ 2.0 k) t_1) (* k t))
     (* -0.3333333333333333 (/ t_1 t)))))

C

double code(double t, double l, double k) {
	double t_1 = pow((l / k), 2.0);
	double tmp;
	if (k <= 2.45e+16) {
		tmp = ((2.0 / k) * t_1) / (k * t);
	} else {
		tmp = -0.3333333333333333 * (t_1 / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) ** 2.0d0
    if (k <= 2.45d+16) then
        tmp = ((2.0d0 / k) * t_1) / (k * t)
    else
        tmp = (-0.3333333333333333d0) * (t_1 / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((l / k), 2.0);
	double tmp;
	if (k <= 2.45e+16) {
		tmp = ((2.0 / k) * t_1) / (k * t);
	} else {
		tmp = -0.3333333333333333 * (t_1 / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((l / k), 2.0)
	tmp = 0
	if k <= 2.45e+16:
		tmp = ((2.0 / k) * t_1) / (k * t)
	else:
		tmp = -0.3333333333333333 * (t_1 / t)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / k) ^ 2.0
	tmp = 0.0
	if (k <= 2.45e+16)
		tmp = Float64(Float64(Float64(2.0 / k) * t_1) / Float64(k * t));
	else
		tmp = Float64(-0.3333333333333333 * Float64(t_1 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) ^ 2.0;
	tmp = 0.0;
	if (k <= 2.45e+16)
		tmp = ((2.0 / k) * t_1) / (k * t);
	else
		tmp = -0.3333333333333333 * (t_1 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 2.45e+16], N[(N[(N[(2.0 / k), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e16

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-/r* 32.4%

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

      4. associate-/r/ 32.4%

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

      5. *-commutative 32.4%

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

      6. times-frac 33.0%

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

      7. +-commutative 33.0%

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

      8. associate--l+ 40.1%

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

      9. metadata-eval 40.1%

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

      10. +-rgt-identity 40.1%

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

      11. times-frac 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 82.6%

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

      1. unpow2 82.6%

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

   6. Simplified 82.6%

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

      1. associate-*l/ 82.6%

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

      2. associate-*l* 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. times-frac 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in k around 0 65.8%

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

       1. unpow2 65.8%

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

       2. unpow2 65.8%

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

   13. Simplified 65.8%

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

       1. associate-*r/ 65.8%

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

       2. times-frac 80.6%

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

       3. pow1 80.6%

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

       4. pow1 80.6%

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

       5. pow-prod-up 80.6%

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

       6. metadata-eval 80.6%

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

   15. Applied egg-rr 80.6%

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

   if 2.45e16 < k

   1. Initial program 42.0%

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

      1. associate-*l* 42.0%

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

      2. associate-*l* 42.0%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 42.0%

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

      7. +-commutative 42.0%

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

      8. associate--l+ 47.0%

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

      9. metadata-eval 47.0%

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

      10. +-rgt-identity 47.0%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 75.6%

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

      1. unpow2 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in k around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. fma-def 54.6%

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

      3. unpow2 54.6%

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

      4. unpow2 54.6%

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

      5. unpow2 54.6%

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

   9. Simplified 54.6%

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

   10. Taylor expanded in k around inf 59.4%

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

       1. associate-/r* 59.4%

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

       2. unpow2 59.4%

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

       3. unpow2 59.4%

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

       4. times-frac 60.9%

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

       5. unpow2 60.9%

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

   12. Simplified 60.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 12: 72.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.45e+16)
   (* (/ 2.0 k) (/ (* l (/ (/ l k) k)) (* k t)))
   (* -0.3333333333333333 (/ (pow (/ l k) 2.0) t))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.45e+16) {
		tmp = (2.0 / k) * ((l * ((l / k) / k)) / (k * t));
	} else {
		tmp = -0.3333333333333333 * (pow((l / k), 2.0) / t);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.45d+16) then
        tmp = (2.0d0 / k) * ((l * ((l / k) / k)) / (k * t))
    else
        tmp = (-0.3333333333333333d0) * (((l / k) ** 2.0d0) / t)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.45e+16) {
		tmp = (2.0 / k) * ((l * ((l / k) / k)) / (k * t));
	} else {
		tmp = -0.3333333333333333 * (Math.pow((l / k), 2.0) / t);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.45e+16:
		tmp = (2.0 / k) * ((l * ((l / k) / k)) / (k * t))
	else:
		tmp = -0.3333333333333333 * (math.pow((l / k), 2.0) / t)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.45e+16)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l * Float64(Float64(l / k) / k)) / Float64(k * t)));
	else
		tmp = Float64(-0.3333333333333333 * Float64((Float64(l / k) ^ 2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.45e+16)
		tmp = (2.0 / k) * ((l * ((l / k) / k)) / (k * t));
	else
		tmp = -0.3333333333333333 * (((l / k) ^ 2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.45e+16], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.45e16

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-/r* 32.4%

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

      4. associate-/r/ 32.4%

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

      5. *-commutative 32.4%

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

      6. times-frac 33.0%

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

      7. +-commutative 33.0%

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

      8. associate--l+ 40.1%

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

      9. metadata-eval 40.1%

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

      10. +-rgt-identity 40.1%

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

      11. times-frac 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in t around 0 82.6%

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

      1. unpow2 82.6%

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

   6. Simplified 82.6%

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

      1. associate-*l/ 82.6%

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

      2. associate-*l* 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. times-frac 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in k around 0 65.8%

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

       1. unpow2 65.8%

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

       2. unpow2 65.8%

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

   13. Simplified 65.8%

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

   14. Taylor expanded in l around 0 65.8%

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

       1. *-rgt-identity 65.8%

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

       2. associate-*r/ 65.8%

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

       3. unpow2 65.8%

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

       4. unpow2 65.8%

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

       5. associate-*l* 76.9%

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

       6. unpow2 76.9%

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

       7. associate-*r/ 76.9%

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

       8. *-rgt-identity 76.9%

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

       9. unpow2 76.9%

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

       10. associate-/r* 79.9%

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

   16. Simplified 79.9%

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

   if 2.45e16 < k

   1. Initial program 42.0%

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

      1. associate-*l* 42.0%

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

      2. associate-*l* 42.0%

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

      3. associate-/r* 42.0%

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

      4. associate-/r/ 42.0%

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

      5. *-commutative 42.0%

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

      6. times-frac 42.0%

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

      7. +-commutative 42.0%

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

      8. associate--l+ 47.0%

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

      9. metadata-eval 47.0%

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

      10. +-rgt-identity 47.0%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 75.6%

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

      1. unpow2 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in k around 0 54.6%

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

      1. *-commutative 54.6%

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

      2. fma-def 54.6%

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

      3. unpow2 54.6%

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

      4. unpow2 54.6%

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

      5. unpow2 54.6%

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

   9. Simplified 54.6%

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

   10. Taylor expanded in k around inf 59.4%

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

       1. associate-/r* 59.4%

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

       2. unpow2 59.4%

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

       3. unpow2 59.4%

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

       4. times-frac 60.9%

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

       5. unpow2 60.9%

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

   12. Simplified 60.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 13: 71.5% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

   1. associate-*l/ 80.9%

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

   2. associate-*l* 86.5%

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

8. Applied egg-rr 86.5%

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

   1. times-frac 87.2%

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

10. Simplified 87.2%

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

11. Taylor expanded in k around 0 63.8%

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

    1. unpow2 63.8%

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

    2. unpow2 63.8%

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

13. Simplified 63.8%

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

14. Taylor expanded in l around 0 63.8%

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

    1. *-rgt-identity 63.8%

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

    2. associate-*r/ 63.8%

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

    3. unpow2 63.8%

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

    4. unpow2 63.8%

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

    5. associate-*l* 72.2%

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

    6. unpow2 72.2%

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

    7. associate-*r/ 72.2%

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

    8. *-rgt-identity 72.2%

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

    9. unpow2 72.2%

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

    10. associate-/r* 74.5%

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

16. Simplified 74.5%

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

17. Final simplification 74.5%

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

Alternative 14: 33.1% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.2%

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

   7. +-commutative 35.2%

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

   8. associate--l+ 41.8%

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

   9. metadata-eval 41.8%

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

   10. +-rgt-identity 41.8%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.8%

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

   1. unpow2 80.8%

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

6. Simplified 80.8%

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

7. Taylor expanded in k around 0 51.9%

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

   1. *-commutative 51.9%

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

   2. fma-def 51.9%

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

   3. unpow2 51.9%

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

   4. unpow2 51.9%

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

   5. unpow2 51.9%

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

9. Simplified 51.9%

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

10. Taylor expanded in k around inf 33.6%

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

    1. associate-/r* 33.7%

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

    2. unpow2 33.7%

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

    3. unpow2 33.7%

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

12. Simplified 33.7%

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

13. Final simplification 33.7%

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

Reproduce

herbie shell --seed 2023252 
(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))))