Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 95.5%

Time: 19.6s

Alternatives: 11

Speedup: 28.1×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% 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: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k \cdot \left(k \cdot t\right)}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{2}{t}}{\sin k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e+152)
   (/ (* l (/ 2.0 (* k (* k t)))) (* (/ (sin k) l) (tan k)))
   (/ (/ (* (pow (/ l k) 2.0) (/ 2.0 t)) (sin k)) (tan k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+152) {
		tmp = (l * (2.0 / (k * (k * t)))) / ((sin(k) / l) * tan(k));
	} else {
		tmp = ((pow((l / k), 2.0) * (2.0 / t)) / sin(k)) / tan(k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d+152) then
        tmp = (l * (2.0d0 / (k * (k * t)))) / ((sin(k) / l) * tan(k))
    else
        tmp = ((((l / k) ** 2.0d0) * (2.0d0 / t)) / sin(k)) / tan(k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+152) {
		tmp = (l * (2.0 / (k * (k * t)))) / ((Math.sin(k) / l) * Math.tan(k));
	} else {
		tmp = ((Math.pow((l / k), 2.0) * (2.0 / t)) / Math.sin(k)) / Math.tan(k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e+152:
		tmp = (l * (2.0 / (k * (k * t)))) / ((math.sin(k) / l) * math.tan(k))
	else:
		tmp = ((math.pow((l / k), 2.0) * (2.0 / t)) / math.sin(k)) / math.tan(k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+152)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(k * Float64(k * t)))) / Float64(Float64(sin(k) / l) * tan(k)));
	else
		tmp = Float64(Float64(Float64((Float64(l / k) ^ 2.0) * Float64(2.0 / t)) / sin(k)) / tan(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+152)
		tmp = (l * (2.0 / (k * (k * t)))) / ((sin(k) / l) * tan(k));
	else
		tmp = ((((l / k) ^ 2.0) * (2.0 / t)) / sin(k)) / tan(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+152], N[(N[(l * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000001e152

   1. Initial program 31.4%

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

      1. associate-*l* 31.4%

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

      2. associate-*l* 31.4%

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

      3. associate-/r* 31.4%

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

      4. associate-/r/ 31.4%

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

      5. *-commutative 31.4%

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

      6. times-frac 32.0%

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

      7. +-commutative 32.0%

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

      8. associate--l+ 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 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in t around 0 85.7%

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

      1. unpow2 85.7%

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

   6. Simplified 85.7%

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

      1. associate-*l/ 85.7%

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

      2. associate-*l* 93.3%

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

   8. Applied egg-rr 93.3%

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

      1. associate-*l/ 93.3%

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

      2. *-commutative 93.3%

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

      3. associate-/r/ 93.4%

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

   10. Simplified 93.4%

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

       1. associate-*l/ 95.8%

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

       2. associate-/r/ 95.8%

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

   12. Applied egg-rr 95.8%

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

   if 2.0000000000000001e152 < (*.f64 l l)

   1. Initial program 33.0%

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

      1. associate-*l* 33.0%

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

      2. associate-*l* 33.0%

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

      3. associate-/r* 33.0%

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

      4. associate-/r/ 33.0%

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

      5. *-commutative 33.0%

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

      6. times-frac 33.1%

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

      7. +-commutative 33.1%

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

      8. associate--l+ 34.9%

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

      9. metadata-eval 34.9%

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

      10. +-rgt-identity 34.9%

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

      11. times-frac 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around 0 64.3%

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

      1. unpow2 64.3%

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

   6. Simplified 64.3%

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

      1. associate-*r/ 64.3%

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

   8. Applied egg-rr 64.3%

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

      1. associate-*r/ 64.3%

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

      2. associate-*r* 66.3%

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

      3. *-commutative 66.3%

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

   10. Applied egg-rr 66.3%

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

   11. Taylor expanded in k around inf 64.3%

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

       1. associate-*r/ 64.3%

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

       2. *-commutative 64.3%

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

       3. associate-*l* 64.3%

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

       4. *-commutative 64.3%

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

       5. associate-/l/ 64.3%

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

       6. *-commutative 64.3%

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

       7. times-frac 68.0%

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

       8. unpow2 68.0%

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

       9. unpow2 68.0%

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

       10. times-frac 94.9%

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

       11. unpow2 94.9%

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

   13. Simplified 94.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k \cdot \left(k \cdot t\right)}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{2}{t}}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 2: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-9)
   (* (/ 2.0 (* k (* k t))) (/ l (/ k (/ l k))))
   (if (<= k 3.2e+150)
     (* (/ 2.0 (* t (* k k))) (* (/ l (tan k)) (/ l (sin k))))
     (* 2.0 (* (pow (/ l k) 2.0) (/ -0.16666666666666666 t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-9) {
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	} else if (k <= 3.2e+150) {
		tmp = (2.0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (-0.16666666666666666 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.2d-9) then
        tmp = (2.0d0 / (k * (k * t))) * (l / (k / (l / k)))
    else if (k <= 3.2d+150) then
        tmp = (2.0d0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)))
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * ((-0.16666666666666666d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-9) {
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	} else if (k <= 3.2e+150) {
		tmp = (2.0 / (t * (k * k))) * ((l / Math.tan(k)) * (l / Math.sin(k)));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (-0.16666666666666666 / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.2e-9:
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)))
	elif k <= 3.2e+150:
		tmp = (2.0 / (t * (k * k))) * ((l / math.tan(k)) * (l / math.sin(k)))
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * (-0.16666666666666666 / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-9)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l / Float64(k / Float64(l / k))));
	elseif (k <= 3.2e+150)
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * k))) * Float64(Float64(l / tan(k)) * Float64(l / sin(k))));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(-0.16666666666666666 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.2e-9)
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	elseif (k <= 3.2e+150)
		tmp = (2.0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)));
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * (-0.16666666666666666 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.2e-9], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.2e+150], N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.2000000000000001e-9

   1. Initial program 34.7%

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

      1. associate-*l* 34.7%

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

      2. associate-*l* 34.7%

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

      3. associate-/r* 34.7%

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

      4. associate-/r/ 34.7%

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

      5. *-commutative 34.7%

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

      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+ 43.4%

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

      9. metadata-eval 43.4%

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

      10. +-rgt-identity 43.4%

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

      11. times-frac 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in t around 0 81.7%

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

      1. unpow2 81.7%

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

   6. Simplified 81.7%

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

      1. associate-*l/ 81.7%

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

      2. associate-*l* 87.8%

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

   8. Applied egg-rr 87.8%

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

      1. associate-*l/ 87.8%

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

      2. *-commutative 87.8%

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

      3. associate-/r/ 87.8%

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

   10. Simplified 87.8%

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

   11. Taylor expanded in k around 0 74.3%

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

       1. unpow2 74.3%

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

       2. associate-/l* 77.8%

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

   13. Simplified 77.8%

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

   if 6.2000000000000001e-9 < k < 3.20000000000000016e150

   1. Initial program 24.6%

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

      1. associate-*l* 24.4%

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

      2. associate-*l* 24.4%

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

      3. associate-/r* 24.5%

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

      4. associate-/r/ 24.4%

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

      5. *-commutative 24.4%

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

      6. times-frac 27.3%

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

      7. +-commutative 27.3%

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

      8. associate--l+ 39.4%

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

      9. metadata-eval 39.4%

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

      10. +-rgt-identity 39.4%

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

      11. times-frac 39.4%

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

   3. Simplified 39.4%

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

   4. Taylor expanded in t around 0 79.1%

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

      1. unpow2 79.1%

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

   6. Simplified 79.1%

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

   if 3.20000000000000016e150 < k

   1. Initial program 25.9%

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

      1. +-rgt-identity 25.7%

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

      2. associate-*l* 25.7%

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

      3. mul0-rgt 10.6%

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

      4. distribute-lft-in 25.9%

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

      5. +-rgt-identity 25.9%

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

      6. sub-neg 25.9%

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

      7. +-commutative 25.9%

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

      8. associate-+l+ 36.4%

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

      9. metadata-eval 36.4%

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

      10. metadata-eval 36.4%

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

      11. +-rgt-identity 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around 0 49.7%

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

      1. unpow2 49.7%

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

      2. times-frac 52.0%

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

      3. unpow2 52.0%

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

      4. *-commutative 52.0%

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

   6. Simplified 52.0%

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

   7. Taylor expanded in k around 0 49.7%

      \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. distribute-lft-out 49.7%

         \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. distribute-rgt-out-- 49.7%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. metadata-eval 49.7%

         \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 49.7%

         \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. times-frac 49.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{k} \cdot \frac{-0.16666666666666666}{k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. unpow2 49.9%

         \[\leadsto 2 \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. associate-/l* 49.9%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 49.9%

         \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 49.9%

         \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

      10. times-frac 51.1%

          \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}\right) \]

   9. Simplified 51.1%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   10. Taylor expanded in k around inf 49.7%

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

       1. associate-*r/ 49.7%

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

       2. *-commutative 49.7%

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

       3. times-frac 49.9%

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

       4. unpow2 49.9%

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

       5. unpow2 49.9%

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

       6. times-frac 56.9%

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

       7. unpow2 56.9%

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

   12. Simplified 56.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \]

Alternative 3: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e-14)
   (* (/ 2.0 (* k (* k t))) (/ l (/ k (/ l k))))
   (if (<= k 3e+150)
     (* (/ 2.0 (* t (* k k))) (/ (* l (/ l (tan k))) (sin k)))
     (* 2.0 (* (pow (/ l k) 2.0) (/ -0.16666666666666666 t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-14) {
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	} else if (k <= 3e+150) {
		tmp = (2.0 / (t * (k * k))) * ((l * (l / tan(k))) / sin(k));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (-0.16666666666666666 / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.95d-14) then
        tmp = (2.0d0 / (k * (k * t))) * (l / (k / (l / k)))
    else if (k <= 3d+150) then
        tmp = (2.0d0 / (t * (k * k))) * ((l * (l / tan(k))) / sin(k))
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * ((-0.16666666666666666d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-14) {
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	} else if (k <= 3e+150) {
		tmp = (2.0 / (t * (k * k))) * ((l * (l / Math.tan(k))) / Math.sin(k));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (-0.16666666666666666 / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.95e-14:
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)))
	elif k <= 3e+150:
		tmp = (2.0 / (t * (k * k))) * ((l * (l / math.tan(k))) / math.sin(k))
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * (-0.16666666666666666 / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e-14)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l / Float64(k / Float64(l / k))));
	elseif (k <= 3e+150)
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * k))) * Float64(Float64(l * Float64(l / tan(k))) / sin(k)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(-0.16666666666666666 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e-14)
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	elseif (k <= 3e+150)
		tmp = (2.0 / (t * (k * k))) * ((l * (l / tan(k))) / sin(k));
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * (-0.16666666666666666 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.95e-14], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3e+150], N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9499999999999999e-14

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-*l* 35.3%

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

      3. associate-/r* 35.3%

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

      4. associate-/r/ 35.3%

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

      5. *-commutative 35.3%

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

      6. times-frac 35.8%

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

      7. +-commutative 35.8%

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

      8. associate--l+ 44.1%

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

      9. metadata-eval 44.1%

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

      10. +-rgt-identity 44.1%

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

      11. times-frac 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in t around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 81.9%

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

      2. associate-*l* 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-/r/ 88.1%

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

   10. Simplified 88.1%

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

   11. Taylor expanded in k around 0 74.3%

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

       1. unpow2 74.3%

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

       2. associate-/l* 78.0%

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

   13. Simplified 78.0%

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

   if 1.9499999999999999e-14 < k < 3.00000000000000012e150

   1. Initial program 22.5%

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

      1. associate-*l* 22.4%

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

      2. associate-*l* 22.4%

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

      3. associate-/r* 22.4%

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

      4. associate-/r/ 22.4%

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

      5. *-commutative 22.4%

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

      6. times-frac 25.0%

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

      7. +-commutative 25.0%

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

      8. associate--l+ 36.1%

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

      9. metadata-eval 36.1%

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

      10. +-rgt-identity 36.1%

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

      11. times-frac 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t around 0 78.1%

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

      1. unpow2 78.1%

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

   6. Simplified 78.1%

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

      1. associate-*l/ 78.1%

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

   8. Applied egg-rr 78.1%

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

   if 3.00000000000000012e150 < k

   1. Initial program 25.9%

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

      1. +-rgt-identity 25.7%

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

      2. associate-*l* 25.7%

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

      3. mul0-rgt 10.6%

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

      4. distribute-lft-in 25.9%

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

      5. +-rgt-identity 25.9%

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

      6. sub-neg 25.9%

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

      7. +-commutative 25.9%

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

      8. associate-+l+ 36.4%

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

      9. metadata-eval 36.4%

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

      10. metadata-eval 36.4%

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

      11. +-rgt-identity 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in t around 0 49.7%

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

      1. unpow2 49.7%

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

      2. times-frac 52.0%

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

      3. unpow2 52.0%

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

      4. *-commutative 52.0%

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

   6. Simplified 52.0%

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

   7. Taylor expanded in k around 0 49.7%

      \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. distribute-lft-out 49.7%

         \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. distribute-rgt-out-- 49.7%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. metadata-eval 49.7%

         \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 49.7%

         \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. times-frac 49.9%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{k} \cdot \frac{-0.16666666666666666}{k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      6. unpow2 49.9%

         \[\leadsto 2 \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      7. associate-/l* 49.9%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      8. unpow2 49.9%

         \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

      9. *-commutative 49.9%

         \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

      10. times-frac 51.1%

          \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}\right) \]

   9. Simplified 51.1%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   10. Taylor expanded in k around inf 49.7%

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

       1. associate-*r/ 49.7%

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

       2. *-commutative 49.7%

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

       3. times-frac 49.9%

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

       4. unpow2 49.9%

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

       5. unpow2 49.9%

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

       6. times-frac 56.9%

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

       7. unpow2 56.9%

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

   12. Simplified 56.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \]

Alternative 4: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{\frac{\frac{\frac{2}{k}}{t}}{k}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e-14)
   (* (/ 2.0 (* k (* k t))) (/ l (/ k (/ l k))))
   (* (/ l (/ (sin k) l)) (/ (/ (/ (/ 2.0 k) t) k) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-14) {
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	} else {
		tmp = (l / (sin(k) / l)) * ((((2.0 / k) / t) / k) / tan(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.95d-14) then
        tmp = (2.0d0 / (k * (k * t))) * (l / (k / (l / k)))
    else
        tmp = (l / (sin(k) / l)) * ((((2.0d0 / k) / t) / k) / tan(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-14) {
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	} else {
		tmp = (l / (Math.sin(k) / l)) * ((((2.0 / k) / t) / k) / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.95e-14:
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)))
	else:
		tmp = (l / (math.sin(k) / l)) * ((((2.0 / k) / t) / k) / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e-14)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l / Float64(k / Float64(l / k))));
	else
		tmp = Float64(Float64(l / Float64(sin(k) / l)) * Float64(Float64(Float64(Float64(2.0 / k) / t) / k) / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e-14)
		tmp = (2.0 / (k * (k * t))) * (l / (k / (l / k)));
	else
		tmp = (l / (sin(k) / l)) * ((((2.0 / k) / t) / k) / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.95e-14], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 / k), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9499999999999999e-14

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-*l* 35.3%

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

      3. associate-/r* 35.3%

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

      4. associate-/r/ 35.3%

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

      5. *-commutative 35.3%

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

      6. times-frac 35.8%

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

      7. +-commutative 35.8%

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

      8. associate--l+ 44.1%

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

      9. metadata-eval 44.1%

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

      10. +-rgt-identity 44.1%

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

      11. times-frac 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in t around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 81.9%

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

      2. associate-*l* 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-/r/ 88.1%

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

   10. Simplified 88.1%

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

   11. Taylor expanded in k around 0 74.3%

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

       1. unpow2 74.3%

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

       2. associate-/l* 78.0%

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

   13. Simplified 78.0%

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

   if 1.9499999999999999e-14 < k

   1. Initial program 24.3%

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

      1. associate-*l* 24.2%

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

      2. associate-*l* 24.2%

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

      3. associate-/r* 24.2%

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

      4. associate-/r/ 24.2%

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

      5. *-commutative 24.2%

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

      6. times-frac 24.3%

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

      7. +-commutative 24.3%

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

      8. associate--l+ 35.2%

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

      9. metadata-eval 35.2%

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

      10. +-rgt-identity 35.2%

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

      11. times-frac 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in t around 0 63.5%

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

      1. unpow2 63.5%

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

   6. Simplified 63.5%

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

      1. associate-*l/ 63.5%

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

      2. associate-*l* 66.4%

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

   8. Applied egg-rr 66.4%

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

      1. associate-*l/ 66.4%

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

      2. *-commutative 66.4%

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

      3. associate-/r/ 66.4%

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

   10. Simplified 66.4%

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

       1. expm1-log1p-u 56.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

       2. expm1-udef 48.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)} - 1} \]

       3. frac-times 50.8%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell \cdot 2}{\frac{\sin k}{\frac{\ell}{\tan k}} \cdot \left(k \cdot \left(k \cdot t\right)\right)}}\right)} - 1 \]

       4. associate-/r/ 50.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot 2}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot \tan k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} - 1 \]

   12. Applied egg-rr 50.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot 2}{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 58.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot 2}{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\right)} \]

       2. expm1-log1p 74.0%

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

       3. times-frac 66.4%

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

       4. associate-*l/ 73.9%

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

       5. times-frac 66.4%

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

       6. associate-/r* 66.7%

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

       7. *-commutative 66.7%

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

       8. associate-/r* 66.8%

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

   14. Simplified 66.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{\frac{\frac{\frac{2}{k}}{t}}{k}}{\tan k}\\ \end{array} \]

Alternative 5: 83.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. associate-*l* 32.1%

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

   2. associate-*l* 32.1%

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

   3. associate-/r* 32.1%

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

   4. associate-/r/ 32.1%

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

   5. *-commutative 32.1%

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

   6. times-frac 32.5%

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

   7. +-commutative 32.5%

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

   8. associate--l+ 41.5%

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

   9. metadata-eval 41.5%

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

   10. +-rgt-identity 41.5%

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

   11. times-frac 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 76.6%

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

   1. unpow2 76.6%

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

6. Simplified 76.6%

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

   1. associate-*l/ 76.6%

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

   2. associate-*l* 81.8%

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

8. Applied egg-rr 81.8%

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

   1. associate-*l/ 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-/r/ 81.8%

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

10. Simplified 81.8%

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

11. Final simplification 81.8%

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

Alternative 6: 83.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. associate-*l* 32.1%

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

   2. associate-*l* 32.1%

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

   3. associate-/r* 32.1%

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

   4. associate-/r/ 32.1%

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

   5. *-commutative 32.1%

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

   6. times-frac 32.5%

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

   7. +-commutative 32.5%

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

   8. associate--l+ 41.5%

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

   9. metadata-eval 41.5%

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

   10. +-rgt-identity 41.5%

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

   11. times-frac 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 76.6%

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

   1. unpow2 76.6%

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

6. Simplified 76.6%

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

   1. associate-*l/ 76.6%

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

   2. associate-*l* 81.8%

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

8. Applied egg-rr 81.8%

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

   1. associate-*l/ 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-/r/ 81.8%

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

10. Simplified 81.8%

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

11. Taylor expanded in k around 0 76.6%

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

    1. unpow2 76.6%

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

    2. associate-*r* 81.8%

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

    3. associate-/r* 82.0%

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

    4. *-commutative 82.0%

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

    5. associate-/r* 82.0%

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

13. Simplified 82.0%

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

14. Final simplification 82.0%

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

Alternative 7: 87.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. associate-*l* 32.1%

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

   2. associate-*l* 32.1%

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

   3. associate-/r* 32.1%

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

   4. associate-/r/ 32.1%

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

   5. *-commutative 32.1%

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

   6. times-frac 32.5%

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

   7. +-commutative 32.5%

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

   8. associate--l+ 41.5%

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

   9. metadata-eval 41.5%

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

   10. +-rgt-identity 41.5%

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

   11. times-frac 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 76.6%

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

   1. unpow2 76.6%

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

6. Simplified 76.6%

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

   1. associate-*l/ 76.6%

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

   2. associate-*l* 81.8%

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

8. Applied egg-rr 81.8%

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

   1. associate-*l/ 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-/r/ 81.8%

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

10. Simplified 81.8%

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

    1. expm1-log1p-u 52.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

    2. expm1-udef 43.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell}{\frac{\sin k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)} - 1} \]

    3. frac-times 44.1%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell \cdot 2}{\frac{\sin k}{\frac{\ell}{\tan k}} \cdot \left(k \cdot \left(k \cdot t\right)\right)}}\right)} - 1 \]

    4. associate-/r/ 44.1%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot 2}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot \tan k\right)} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} - 1 \]

12. Applied egg-rr 44.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot 2}{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 54.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot 2}{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\right)} \]

    2. expm1-log1p 85.2%

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

    3. associate-/l* 85.2%

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

    4. associate-*l* 84.5%

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

14. Simplified 84.5%

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

15. Final simplification 84.5%

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

Alternative 8: 89.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. associate-*l* 32.1%

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

   2. associate-*l* 32.1%

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

   3. associate-/r* 32.1%

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

   4. associate-/r/ 32.1%

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

   5. *-commutative 32.1%

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

   6. times-frac 32.5%

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

   7. +-commutative 32.5%

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

   8. associate--l+ 41.5%

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

   9. metadata-eval 41.5%

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

   10. +-rgt-identity 41.5%

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

   11. times-frac 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 76.6%

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

   1. unpow2 76.6%

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

6. Simplified 76.6%

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

   1. associate-*l/ 76.6%

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

   2. associate-*l* 81.8%

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

8. Applied egg-rr 81.8%

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

   1. associate-*l/ 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-/r/ 81.8%

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

10. Simplified 81.8%

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

    1. associate-*l/ 86.3%

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

    2. associate-/r/ 86.3%

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

12. Applied egg-rr 86.3%

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

13. Final simplification 86.3%

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

Alternative 9: 69.5% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. associate-*l* 32.1%

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

   2. associate-*l* 32.1%

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

   3. associate-/r* 32.1%

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

   4. associate-/r/ 32.1%

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

   5. *-commutative 32.1%

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

   6. times-frac 32.5%

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

   7. +-commutative 32.5%

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

   8. associate--l+ 41.5%

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

   9. metadata-eval 41.5%

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

   10. +-rgt-identity 41.5%

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

   11. times-frac 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 76.6%

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

   1. unpow2 76.6%

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

6. Simplified 76.6%

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

7. Taylor expanded in k around 0 61.2%

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

   1. unpow2 61.2%

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

   2. unpow2 61.2%

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

   3. times-frac 67.1%

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

9. Simplified 67.1%

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

10. Final simplification 67.1%

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

Alternative 10: 70.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. associate-*l* 32.1%

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

   2. associate-*l* 32.1%

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

   3. associate-/r* 32.1%

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

   4. associate-/r/ 32.1%

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

   5. *-commutative 32.1%

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

   6. times-frac 32.5%

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

   7. +-commutative 32.5%

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

   8. associate--l+ 41.5%

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

   9. metadata-eval 41.5%

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

   10. +-rgt-identity 41.5%

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

   11. times-frac 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 76.6%

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

   1. unpow2 76.6%

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

6. Simplified 76.6%

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

   1. associate-*l/ 76.6%

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

   2. associate-*l* 81.8%

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

8. Applied egg-rr 81.8%

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

   1. associate-*l/ 81.8%

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

   2. *-commutative 81.8%

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

   3. associate-/r/ 81.8%

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

10. Simplified 81.8%

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

11. Taylor expanded in k around 0 67.1%

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

    1. unpow2 67.1%

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

    2. associate-/l* 69.7%

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

13. Simplified 69.7%

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

14. Final simplification 69.7%

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

Alternative 11: 32.8% accurate, 32.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.1%

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

   1. +-rgt-identity 19.1%

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

   2. associate-*l* 19.1%

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

   3. mul0-rgt 18.1%

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

   4. distribute-lft-in 27.8%

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

   5. +-rgt-identity 32.1%

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

   6. sub-neg 32.1%

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

   7. +-commutative 32.1%

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

   8. associate-+l+ 41.1%

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

   9. metadata-eval 41.1%

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

   10. metadata-eval 41.1%

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

   11. +-rgt-identity 41.1%

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

3. Simplified 41.1%

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

4. Taylor expanded in t around 0 70.4%

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

   1. unpow2 70.4%

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

   2. times-frac 71.3%

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

   3. unpow2 71.3%

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

   4. *-commutative 71.3%

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

6. Simplified 71.3%

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

7. Taylor expanded in k around 0 33.0%

   \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. distribute-rgt-out-- 33.0%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. metadata-eval 33.0%

      \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 33.0%

      \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. times-frac 35.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{k} \cdot \frac{-0.16666666666666666}{k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   6. unpow2 35.0%

      \[\leadsto 2 \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   7. associate-/l* 35.1%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   8. unpow2 35.1%

      \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   9. *-commutative 35.1%

      \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   10. times-frac 40.4%

       \[\leadsto 2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}\right) \]

9. Simplified 40.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Taylor expanded in k around inf 27.1%

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

    1. associate-*r/ 27.1%

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

    2. unpow2 27.1%

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

    3. unpow2 27.1%

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

    4. associate-*r* 27.7%

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

12. Simplified 27.7%

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

13. Final simplification 27.7%

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

Reproduce

herbie shell --seed 2023230 
(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))))