Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 96.7%

Time: 19.3s

Alternatives: 8

Speedup: 28.1×

• 35.0% 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: 34.5% 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: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+232}:\\ \;\;\;\;\frac{\frac{2}{k \cdot t}}{\frac{\sin k}{\frac{\ell}{k}}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+232)
   (* (/ (/ 2.0 (* k t)) (/ (sin k) (/ l k))) (/ l (tan k)))
   (* 2.0 (/ (/ (cos k) (* (/ k l) (/ k l))) (* t (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+232) {
		tmp = ((2.0 / (k * t)) / (sin(k) / (l / k))) * (l / tan(k));
	} else {
		tmp = 2.0 * ((cos(k) / ((k / l) * (k / l))) / (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) <= 1d+232) then
        tmp = ((2.0d0 / (k * t)) / (sin(k) / (l / k))) * (l / tan(k))
    else
        tmp = 2.0d0 * ((cos(k) / ((k / l) * (k / l))) / (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) <= 1e+232) {
		tmp = ((2.0 / (k * t)) / (Math.sin(k) / (l / k))) * (l / Math.tan(k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / ((k / l) * (k / l))) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+232:
		tmp = ((2.0 / (k * t)) / (math.sin(k) / (l / k))) * (l / math.tan(k))
	else:
		tmp = 2.0 * ((math.cos(k) / ((k / l) * (k / l))) / (t * math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+232)
		tmp = Float64(Float64(Float64(2.0 / Float64(k * t)) / Float64(sin(k) / Float64(l / k))) * Float64(l / tan(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(Float64(k / l) * Float64(k / l))) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+232)
		tmp = ((2.0 / (k * t)) / (sin(k) / (l / k))) * (l / tan(k));
	else
		tmp = 2.0 * ((cos(k) / ((k / l) * (k / l))) / (t * (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+232], N[(N[(N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000006e232

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*l* 36.0%

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

      3. associate-/r* 36.0%

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

      4. associate-/r/ 35.4%

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

      5. *-commutative 35.4%

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

      6. times-frac 36.5%

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

      7. +-commutative 36.5%

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

      8. associate--l+ 44.2%

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

      9. metadata-eval 44.2%

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

      10. +-rgt-identity 44.2%

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

      11. times-frac 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in t around 0 86.2%

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

      1. unpow2 86.2%

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

   6. Simplified 86.2%

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

      1. associate-*l/ 86.6%

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

      2. associate-*l* 90.2%

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

   8. Applied egg-rr 90.2%

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

      1. associate-*l/ 89.8%

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

      2. associate-*r* 93.7%

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

      3. associate-*r/ 93.7%

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

      4. associate-*r* 90.1%

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

      5. unpow2 90.1%

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

      6. *-commutative 90.1%

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

      7. associate-/r* 90.5%

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

      8. unpow2 90.5%

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

   10. Simplified 90.5%

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

       1. associate-*l/ 88.7%

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

       2. associate-*l/ 91.0%

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

   12. Applied egg-rr 91.0%

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

       1. frac-times 96.7%

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

       2. associate-*l/ 98.4%

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

       3. associate-/l* 98.5%

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

       4. associate-/l/ 98.5%

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

   14. Applied egg-rr 98.5%

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

   if 1.00000000000000006e232 < (*.f64 l l)

   1. Initial program 38.7%

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

      1. associate-*l* 38.7%

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

      2. associate-*l* 38.8%

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

      3. associate-/r* 38.7%

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

      4. associate-/r/ 38.8%

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

      5. *-commutative 38.8%

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

      6. times-frac 36.4%

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

      7. +-commutative 36.4%

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

      8. associate--l+ 37.7%

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

      9. metadata-eval 37.7%

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

      10. +-rgt-identity 37.7%

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

      11. times-frac 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around 0 54.0%

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

      1. unpow2 54.0%

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

   6. Simplified 54.0%

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

      1. associate-*l/ 54.0%

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

      2. associate-*l* 55.3%

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

   8. Applied egg-rr 55.3%

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

      1. associate-*l/ 55.3%

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

      2. associate-*r* 70.2%

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

      3. associate-*r/ 70.2%

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

      4. associate-*r* 66.6%

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

      5. unpow2 66.6%

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

      6. *-commutative 66.6%

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

      7. associate-/r* 66.6%

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

      8. unpow2 66.6%

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

   10. Simplified 66.6%

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

       1. frac-times 66.6%

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

       2. associate-*l/ 68.8%

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

   12. Applied egg-rr 68.8%

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

       1. times-frac 68.8%

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

       2. times-frac 87.4%

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

   14. Simplified 87.4%

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

   15. Taylor expanded in t around 0 53.9%

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

       1. unpow2 53.9%

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

       2. associate-/r* 54.0%

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

       3. associate-/l* 54.0%

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

       4. unpow2 54.0%

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

       5. times-frac 93.7%

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

   17. Simplified 93.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+232}:\\ \;\;\;\;\frac{\frac{2}{k \cdot t}}{\frac{\sin k}{\frac{\ell}{k}}} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 2: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* (/ l k) (/ (* (/ l k) (/ (/ 2.0 t) k)) (sin k)))
   (* (/ 2.0 (* t (* k k))) (* (/ l (tan k)) (/ l (sin k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / sin(k));
	} else {
		tmp = (2.0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = (l / k) * (((l / k) * ((2.0d0 / t) / k)) / sin(k))
    else
        tmp = (2.0d0 / (t * (k * k))) * ((l / tan(k)) * (l / sin(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / Math.sin(k));
	} else {
		tmp = (2.0 / (t * (k * k))) * ((l / Math.tan(k)) * (l / Math.sin(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / math.sin(k))
	else:
		tmp = (2.0 / (t * (k * k))) * ((l / math.tan(k)) * (l / math.sin(k)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. associate-*l* 24.4%

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

      3. associate-/r* 24.4%

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

      4. associate-/r/ 22.9%

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

      5. *-commutative 22.9%

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

      6. times-frac 22.9%

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

      7. +-commutative 22.9%

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

      8. associate--l+ 35.1%

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

      9. metadata-eval 35.1%

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

      10. +-rgt-identity 35.1%

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

      11. times-frac 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in t around 0 83.9%

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

      1. unpow2 83.9%

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

   6. Simplified 83.9%

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

      1. associate-*l/ 83.9%

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

      2. associate-*l* 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. associate-*l/ 85.9%

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

      2. associate-*r* 94.3%

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

      3. associate-*r/ 94.3%

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

      4. associate-*r* 92.4%

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

      5. unpow2 92.4%

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

      6. *-commutative 92.4%

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

      7. associate-/r* 92.4%

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

      8. unpow2 92.4%

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

   10. Simplified 92.4%

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

       1. frac-times 84.4%

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

       2. associate-*l/ 88.5%

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

   12. Applied egg-rr 88.5%

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

       1. times-frac 88.7%

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

       2. times-frac 97.2%

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

   14. Simplified 97.2%

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

   15. Taylor expanded in k around 0 97.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.2%

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

      1. associate-*l* 41.2%

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

      2. associate-*l* 41.2%

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

      3. associate-/r* 41.2%

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

      4. associate-/r/ 41.2%

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

      5. *-commutative 41.2%

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

      6. times-frac 41.2%

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

      7. +-commutative 41.2%

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

      8. associate--l+ 44.6%

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

      9. metadata-eval 44.6%

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

      10. +-rgt-identity 44.6%

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

      11. times-frac 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in t around 0 73.3%

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

      1. unpow2 73.3%

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

   6. Simplified 73.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \]

Alternative 3: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* (/ l k) (/ (* (/ l k) (/ (/ 2.0 t) k)) (sin k)))
   (* (/ (* l (/ l (tan k))) (sin k)) (/ 2.0 (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / sin(k));
	} else {
		tmp = ((l * (l / tan(k))) / sin(k)) * (2.0 / (k * (k * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = (l / k) * (((l / k) * ((2.0d0 / t) / k)) / sin(k))
    else
        tmp = ((l * (l / tan(k))) / sin(k)) * (2.0d0 / (k * (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / Math.sin(k));
	} else {
		tmp = ((l * (l / Math.tan(k))) / Math.sin(k)) * (2.0 / (k * (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / math.sin(k))
	else:
		tmp = ((l * (l / math.tan(k))) / math.sin(k)) * (2.0 / (k * (k * t)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. associate-*l* 24.4%

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

      3. associate-/r* 24.4%

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

      4. associate-/r/ 22.9%

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

      5. *-commutative 22.9%

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

      6. times-frac 22.9%

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

      7. +-commutative 22.9%

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

      8. associate--l+ 35.1%

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

      9. metadata-eval 35.1%

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

      10. +-rgt-identity 35.1%

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

      11. times-frac 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in t around 0 83.9%

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

      1. unpow2 83.9%

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

   6. Simplified 83.9%

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

      1. associate-*l/ 83.9%

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

      2. associate-*l* 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. associate-*l/ 85.9%

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

      2. associate-*r* 94.3%

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

      3. associate-*r/ 94.3%

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

      4. associate-*r* 92.4%

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

      5. unpow2 92.4%

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

      6. *-commutative 92.4%

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

      7. associate-/r* 92.4%

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

      8. unpow2 92.4%

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

   10. Simplified 92.4%

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

       1. frac-times 84.4%

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

       2. associate-*l/ 88.5%

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

   12. Applied egg-rr 88.5%

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

       1. times-frac 88.7%

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

       2. times-frac 97.2%

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

   14. Simplified 97.2%

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

   15. Taylor expanded in k around 0 97.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.2%

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

      1. associate-*l* 41.2%

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

      2. associate-*l* 41.2%

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

      3. associate-/r* 41.2%

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

      4. associate-/r/ 41.2%

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

      5. *-commutative 41.2%

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

      6. times-frac 41.2%

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

      7. +-commutative 41.2%

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

      8. associate--l+ 44.6%

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

      9. metadata-eval 44.6%

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

      10. +-rgt-identity 44.6%

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

      11. times-frac 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in t around 0 73.3%

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

      1. unpow2 73.3%

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

   6. Simplified 73.3%

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

      1. associate-*l/ 73.6%

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

      2. associate-*l* 76.8%

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

   8. Applied egg-rr 76.8%

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

      1. associate-*l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*l/ 76.5%

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

   10. Simplified 76.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 4: 94.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.9%

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

   2. associate-*l* 36.9%

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

   3. associate-/r* 36.9%

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

   4. associate-/r/ 36.5%

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

   5. *-commutative 36.5%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.2%

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

   9. metadata-eval 42.2%

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

   10. +-rgt-identity 42.2%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 76.0%

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

   1. unpow2 76.0%

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

6. Simplified 76.0%

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

   1. associate-*l/ 76.2%

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

   2. associate-*l* 79.1%

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

8. Applied egg-rr 79.1%

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

   1. associate-*l/ 78.9%

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

   2. associate-*r* 86.2%

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

   3. associate-*r/ 86.2%

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

   4. associate-*r* 82.7%

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

   5. unpow2 82.7%

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

   6. *-commutative 82.7%

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

   7. associate-/r* 82.9%

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

   8. unpow2 82.9%

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

10. Simplified 82.9%

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

    1. frac-times 80.5%

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

    2. associate-*l/ 83.4%

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

12. Applied egg-rr 83.4%

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

    1. times-frac 83.8%

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

    2. times-frac 95.0%

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

14. Simplified 95.0%

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

15. Final simplification 95.0%

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

Alternative 5: 75.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-29}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-29)
   (* (/ l k) (/ (* (/ l k) (/ (/ 2.0 t) k)) (sin k)))
   (/ 2.0 (* (/ (* k k) (cos k)) (* (/ t l) (/ k (/ l k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-29) {
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / sin(k));
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * (k / (l / 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) <= 1d-29) then
        tmp = (l / k) * (((l / k) * ((2.0d0 / t) / k)) / sin(k))
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((t / l) * (k / (l / k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-29) {
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / Math.sin(k));
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((t / l) * (k / (l / k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-29:
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / math.sin(k))
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((t / l) * (k / (l / k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-29)
		tmp = Float64(Float64(l / k) * Float64(Float64(Float64(l / k) * Float64(Float64(2.0 / t) / k)) / sin(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(t / l) * Float64(k / Float64(l / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-29)
		tmp = (l / k) * (((l / k) * ((2.0 / t) / k)) / sin(k));
	else
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * (k / (l / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-29], N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999943e-30

   1. Initial program 32.8%

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

      1. associate-*l* 32.8%

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

      2. associate-*l* 32.8%

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

      3. associate-/r* 32.8%

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

      4. associate-/r/ 32.0%

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

      5. *-commutative 32.0%

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

      6. times-frac 32.9%

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

      7. +-commutative 32.9%

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

      8. associate--l+ 42.8%

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

      9. metadata-eval 42.8%

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

      10. +-rgt-identity 42.8%

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

      11. times-frac 52.1%

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

   3. Simplified 52.1%

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

   4. Taylor expanded in t around 0 89.1%

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

      1. unpow2 89.1%

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

   6. Simplified 89.1%

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

      1. associate-*l/ 89.6%

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

      2. associate-*l* 91.7%

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

   8. Applied egg-rr 91.7%

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

      1. associate-*l/ 91.2%

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

      2. associate-*r* 96.1%

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

      3. associate-*r/ 96.0%

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

      4. associate-*r* 94.0%

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

      5. unpow2 94.0%

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

      6. *-commutative 94.0%

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

      7. associate-/r* 93.9%

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

      8. unpow2 93.9%

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

   10. Simplified 93.9%

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

       1. frac-times 88.8%

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

       2. associate-*l/ 90.9%

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

   12. Applied egg-rr 90.9%

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

       1. times-frac 91.7%

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

       2. times-frac 98.3%

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

   14. Simplified 98.3%

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

   15. Taylor expanded in k around 0 88.2%

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

   if 9.99999999999999943e-30 < (*.f64 l l)

   1. Initial program 40.5%

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

      1. +-rgt-identity 20.3%

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

      2. associate-*l* 20.3%

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

      3. mul0-rgt 26.2%

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

      4. distribute-lft-in 35.3%

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

      5. +-rgt-identity 40.5%

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

      6. sub-neg 40.5%

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

      7. +-commutative 40.5%

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

      8. associate-+l+ 42.2%

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

      9. metadata-eval 42.2%

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

      10. metadata-eval 42.2%

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

      11. +-rgt-identity 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in t around 0 65.0%

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

      1. unpow2 65.0%

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

      2. times-frac 66.8%

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

      3. unpow2 66.8%

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

      4. *-commutative 66.8%

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

   6. Simplified 66.8%

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

      1. times-frac 71.2%

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

   8. Applied egg-rr 71.2%

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

   9. Taylor expanded in k around 0 57.4%

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

       1. unpow2 57.4%

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

       2. associate-/l* 57.5%

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

   11. Simplified 57.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-29}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{2}{t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \end{array} \]

Alternative 6: 73.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-*l* 36.9%

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

   2. associate-*l* 36.9%

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

   3. associate-/r* 36.9%

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

   4. associate-/r/ 36.5%

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

   5. *-commutative 36.5%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.2%

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

   9. metadata-eval 42.2%

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

   10. +-rgt-identity 42.2%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 76.0%

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

   1. unpow2 76.0%

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

6. Simplified 76.0%

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

   1. associate-*l/ 76.2%

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

   2. associate-*l* 79.1%

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

8. Applied egg-rr 79.1%

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

   1. associate-*l/ 78.9%

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

   2. associate-*r* 86.2%

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

   3. associate-*r/ 86.2%

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

   4. associate-*r* 82.7%

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

   5. unpow2 82.7%

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

   6. *-commutative 82.7%

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

   7. associate-/r* 82.9%

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

   8. unpow2 82.9%

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

10. Simplified 82.9%

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

    1. frac-times 80.5%

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

    2. associate-*l/ 83.4%

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

12. Applied egg-rr 83.4%

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

    1. times-frac 83.8%

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

    2. times-frac 95.0%

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

14. Simplified 95.0%

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

15. Taylor expanded in k around 0 69.4%

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

16. Final simplification 69.4%

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

Alternative 7: 70.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+177}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell \cdot \frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+177)
   (/ 2.0 (* t (* (/ k (/ l k)) (/ (* k k) l))))
   (/ 2.0 (/ (pow k 4.0) (* l (/ l t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+177) {
		tmp = 2.0 / (t * ((k / (l / k)) * ((k * k) / l)));
	} else {
		tmp = 2.0 / (pow(k, 4.0) / (l * (l / 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 ((l * l) <= 1d+177) then
        tmp = 2.0d0 / (t * ((k / (l / k)) * ((k * k) / l)))
    else
        tmp = 2.0d0 / ((k ** 4.0d0) / (l * (l / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+177) {
		tmp = 2.0 / (t * ((k / (l / k)) * ((k * k) / l)));
	} else {
		tmp = 2.0 / (Math.pow(k, 4.0) / (l * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+177:
		tmp = 2.0 / (t * ((k / (l / k)) * ((k * k) / l)))
	else:
		tmp = 2.0 / (math.pow(k, 4.0) / (l * (l / t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+177)
		tmp = Float64(2.0 / Float64(t * Float64(Float64(k / Float64(l / k)) * Float64(Float64(k * k) / l))));
	else
		tmp = Float64(2.0 / Float64((k ^ 4.0) / Float64(l * Float64(l / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+177)
		tmp = 2.0 / (t * ((k / (l / k)) * ((k * k) / l)));
	else
		tmp = 2.0 / ((k ^ 4.0) / (l * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+177], N[(2.0 / N[(t * N[(N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e177

   1. Initial program 35.7%

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

      1. +-rgt-identity 29.0%

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

      2. associate-*l* 29.0%

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

      3. mul0-rgt 18.6%

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

      4. distribute-lft-in 34.5%

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

      5. +-rgt-identity 35.7%

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

      6. sub-neg 35.7%

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

      7. +-commutative 35.7%

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

      8. associate-+l+ 43.2%

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

      9. metadata-eval 43.2%

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

      10. metadata-eval 43.2%

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

      11. +-rgt-identity 43.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in k around 0 65.4%

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

      1. unpow2 65.4%

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

      2. associate-/l* 63.0%

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

      3. associate-/r/ 66.0%

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

   6. Simplified 66.0%

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

      1. sqr-pow 66.0%

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

      2. metadata-eval 66.0%

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

      3. pow2 66.0%

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

      4. metadata-eval 66.0%

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

      5. pow2 66.0%

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

   8. Applied egg-rr 66.0%

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

      1. times-frac 79.7%

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

   10. Applied egg-rr 79.7%

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

   11. Taylor expanded in k around 0 79.7%

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

       1. unpow2 79.7%

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

       2. associate-/l* 79.7%

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

   13. Simplified 79.7%

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

   if 1e177 < (*.f64 l l)

   1. Initial program 39.0%

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

      1. +-rgt-identity 17.9%

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

      2. associate-*l* 17.9%

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

      3. mul0-rgt 23.4%

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

      4. distribute-lft-in 32.4%

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

      5. +-rgt-identity 39.0%

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

      6. sub-neg 39.0%

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

      7. +-commutative 39.0%

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

      8. associate-+l+ 40.1%

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

      9. metadata-eval 40.1%

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

      10. metadata-eval 40.1%

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

      11. +-rgt-identity 40.1%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in k around 0 44.5%

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

      1. unpow2 44.5%

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

      2. associate-/l* 45.5%

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

      3. associate-/r/ 44.0%

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

   6. Simplified 44.0%

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

      1. sqr-pow 44.0%

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

      2. metadata-eval 44.0%

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

      3. pow2 44.0%

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

      4. metadata-eval 44.0%

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

      5. pow2 44.0%

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

   8. Applied egg-rr 44.0%

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

   9. Taylor expanded in k around 0 44.5%

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

       1. associate-/l* 45.5%

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

       2. unpow2 45.5%

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

       3. associate-*r/ 48.3%

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

   11. Simplified 48.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\ell \cdot \frac{\ell}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+177}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell \cdot \frac{\ell}{t}}}\\ \end{array} \]

Alternative 8: 69.9% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. +-rgt-identity 25.0%

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

   2. associate-*l* 25.0%

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

   3. mul0-rgt 20.3%

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

   4. distribute-lft-in 33.7%

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

   5. +-rgt-identity 36.9%

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

   6. sub-neg 36.9%

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

   7. +-commutative 36.9%

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

   8. associate-+l+ 42.1%

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

   9. metadata-eval 42.1%

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

   10. metadata-eval 42.1%

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

   11. +-rgt-identity 42.1%

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

3. Simplified 42.1%

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

4. Taylor expanded in k around 0 58.0%

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

   1. unpow2 58.0%

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

   2. associate-/l* 56.8%

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

   3. associate-/r/ 58.2%

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

6. Simplified 58.2%

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

   1. sqr-pow 58.2%

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

   2. metadata-eval 58.2%

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

   3. pow2 58.2%

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

   4. metadata-eval 58.2%

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

   5. pow2 58.2%

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

8. Applied egg-rr 58.2%

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

   1. times-frac 66.7%

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

10. Applied egg-rr 66.7%

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

11. Taylor expanded in k around 0 66.7%

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

    1. unpow2 66.7%

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

    2. associate-/l* 66.7%

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

13. Simplified 66.7%

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

14. Final simplification 66.7%

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

Reproduce

herbie shell --seed 2023178 
(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))))