Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 95.5%

Time: 17.5s

Alternatives: 4

Speedup: 28.1×

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{k}}{k \cdot t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+122)
   (* (* l (/ (/ 2.0 k) (* k t))) (/ (/ l (sin k)) (tan k)))
   (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+122) {
		tmp = (l * ((2.0 / k) / (k * t))) * ((l / sin(k)) / tan(k));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	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) <= 5d+122) then
        tmp = (l * ((2.0d0 / k) / (k * t))) * ((l / sin(k)) / tan(k))
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+122) {
		tmp = (l * ((2.0 / k) / (k * t))) * ((l / Math.sin(k)) / Math.tan(k));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e+122:
		tmp = (l * ((2.0 / k) / (k * t))) * ((l / math.sin(k)) / math.tan(k))
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+122)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / k) / Float64(k * t))) * Float64(Float64(l / sin(k)) / tan(k)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e+122)
		tmp = (l * ((2.0 / k) / (k * t))) * ((l / sin(k)) / tan(k));
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+122], N[(N[(l * N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999989e122

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-*l* 34.4%

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

      3. associate-/r* 33.8%

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

      4. associate-/r/ 33.3%

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

      5. *-commutative 33.3%

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

      6. times-frac 34.6%

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

      7. +-commutative 34.6%

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

      8. associate--l+ 46.7%

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

      9. metadata-eval 46.7%

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

      10. +-rgt-identity 46.7%

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

      11. times-frac 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in t around 0 90.4%

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

      1. unpow2 90.4%

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

   6. Simplified 90.4%

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

      1. associate-*l/ 90.4%

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

      2. associate-*l* 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. associate-*l/ 94.5%

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

      2. *-commutative 94.5%

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

      3. associate-*l* 96.8%

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

   10. Simplified 96.8%

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

       1. associate-*l/ 96.8%

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

   12. Applied egg-rr 96.8%

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

       1. expm1-log1p-u 80.5%

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

       2. expm1-udef 63.6%

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

       3. associate-/l* 63.6%

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

   14. Applied egg-rr 63.6%

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

       1. expm1-def 80.5%

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

       2. expm1-log1p 95.1%

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

       3. associate-/r/ 96.8%

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

       4. times-frac 95.1%

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

       5. *-commutative 95.1%

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

       6. associate-*r/ 90.9%

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

       7. *-commutative 90.9%

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

       8. times-frac 96.9%

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

       9. associate-*r/ 96.9%

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

       10. associate-/r* 96.8%

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

   16. Simplified 96.8%

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

   if 4.99999999999999989e122 < (*.f64 l l)

   1. Initial program 31.5%

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

      1. associate-*l* 31.5%

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

      2. associate-*l* 31.5%

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

      3. associate-/r* 31.5%

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

      4. associate-/r/ 31.5%

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

      5. *-commutative 31.5%

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

      6. times-frac 31.7%

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

      7. +-commutative 31.7%

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

      8. associate--l+ 32.8%

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

      9. metadata-eval 32.8%

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

      10. +-rgt-identity 32.8%

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

      11. times-frac 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around 0 56.7%

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

      1. unpow2 56.7%

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

   6. Simplified 56.7%

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

      1. associate-*l/ 56.7%

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

      2. associate-*l* 60.1%

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

   8. Applied egg-rr 60.1%

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

      1. associate-*l/ 60.1%

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

      2. *-commutative 60.1%

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

      3. associate-*l* 73.1%

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

   10. Simplified 73.1%

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

       1. associate-*l/ 73.1%

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

   12. Applied egg-rr 73.1%

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

   13. Taylor expanded in l around 0 57.6%

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

       1. *-commutative 57.6%

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

       2. times-frac 56.9%

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

       3. unpow2 56.9%

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

       4. unpow2 56.9%

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

       5. times-frac 94.5%

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

       6. unpow2 94.5%

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

       7. *-commutative 94.5%

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

   15. Simplified 94.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{k}}{k \cdot t}\right) \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 89.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. associate-*l* 33.4%

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

   2. associate-*l* 33.4%

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

   3. associate-/r* 33.0%

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

   4. associate-/r/ 32.7%

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

   5. *-commutative 32.7%

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

   6. times-frac 33.5%

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

   7. +-commutative 33.5%

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

   8. associate--l+ 41.7%

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

   9. metadata-eval 41.7%

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

   10. +-rgt-identity 41.7%

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

   11. times-frac 46.8%

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

3. Simplified 46.8%

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

4. Taylor expanded in t around 0 78.2%

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

   1. unpow2 78.2%

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

6. Simplified 78.2%

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

   1. associate-*l/ 78.2%

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

   2. associate-*l* 82.0%

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

8. Applied egg-rr 82.0%

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

   1. associate-*l/ 82.0%

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

   2. *-commutative 82.0%

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

   3. associate-*l* 88.2%

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

10. Simplified 88.2%

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

    1. associate-*l/ 88.2%

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

12. Applied egg-rr 88.2%

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

    1. expm1-log1p-u 65.0%

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

    2. expm1-udef 53.1%

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

    3. associate-/l* 53.1%

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

14. Applied egg-rr 53.1%

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

    1. expm1-def 65.0%

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

    2. expm1-log1p 87.1%

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

    3. associate-/r/ 88.2%

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

    4. times-frac 87.1%

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

    5. *-commutative 87.1%

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

    6. associate-*r/ 79.7%

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

    7. *-commutative 79.7%

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

    8. times-frac 88.2%

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

    9. associate-*r/ 88.2%

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

    10. associate-/r* 89.0%

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

16. Simplified 89.0%

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

17. Final simplification 89.0%

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

Alternative 3: 71.0% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. associate-*l* 33.4%

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

   2. associate-*l* 33.4%

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

   3. associate-/r* 33.0%

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

   4. associate-/r/ 32.7%

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

   5. *-commutative 32.7%

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

   6. times-frac 33.5%

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

   7. +-commutative 33.5%

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

   8. associate--l+ 41.7%

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

   9. metadata-eval 41.7%

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

   10. +-rgt-identity 41.7%

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

   11. times-frac 46.8%

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

3. Simplified 46.8%

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

4. Taylor expanded in t around 0 78.2%

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

   1. unpow2 78.2%

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

6. Simplified 78.2%

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

   1. associate-*l/ 78.2%

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

   2. associate-*l* 82.0%

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

8. Applied egg-rr 82.0%

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

   1. associate-*l/ 82.0%

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

   2. *-commutative 82.0%

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

   3. associate-*l* 88.2%

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

10. Simplified 88.2%

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

11. Taylor expanded in k around 0 69.3%

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

12. Taylor expanded in k around 0 70.1%

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

13. Final simplification 70.1%

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

Alternative 4: 69.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. associate-*l* 33.4%

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

   2. associate-*l* 33.4%

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

   3. associate-/r* 33.0%

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

   4. associate-/r/ 32.7%

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

   5. *-commutative 32.7%

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

   6. times-frac 33.5%

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

   7. +-commutative 33.5%

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

   8. associate--l+ 41.7%

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

   9. metadata-eval 41.7%

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

   10. +-rgt-identity 41.7%

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

   11. times-frac 46.8%

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

3. Simplified 46.8%

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

4. Taylor expanded in t around 0 78.2%

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

   1. unpow2 78.2%

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

6. Simplified 78.2%

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

   1. associate-*l/ 78.2%

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

   2. associate-*l* 82.0%

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

8. Applied egg-rr 82.0%

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

9. Taylor expanded in k around 0 61.5%

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

    1. unpow2 61.5%

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

    2. unpow2 61.5%

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

    3. times-frac 70.5%

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

11. Simplified 70.5%

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

12. Final simplification 70.5%

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

Reproduce

herbie shell --seed 2023257 
(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))))