Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.1% → 93.3%

Time: 20.7s

Alternatives: 12

Speedup: 28.1×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.1% 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: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-237)
   (* (/ l (sin k)) (* (/ l (tan k)) (/ 2.0 (* k (* k t)))))
   (* 2.0 (/ (cos k) (/ (* t (pow (sin k) 2.0)) (pow (/ l k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-237) {
		tmp = (l / sin(k)) * ((l / tan(k)) * (2.0 / (k * (k * t))));
	} else {
		tmp = 2.0 * (cos(k) / ((t * pow(sin(k), 2.0)) / pow((l / k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-237) then
        tmp = (l / sin(k)) * ((l / tan(k)) * (2.0d0 / (k * (k * t))))
    else
        tmp = 2.0d0 * (cos(k) / ((t * (sin(k) ** 2.0d0)) / ((l / k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-237) {
		tmp = (l / Math.sin(k)) * ((l / Math.tan(k)) * (2.0 / (k * (k * t))));
	} else {
		tmp = 2.0 * (Math.cos(k) / ((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-237:
		tmp = (l / math.sin(k)) * ((l / math.tan(k)) * (2.0 / (k * (k * t))))
	else:
		tmp = 2.0 * (math.cos(k) / ((t * math.pow(math.sin(k), 2.0)) / math.pow((l / k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-237)
		tmp = Float64(Float64(l / sin(k)) * Float64(Float64(l / tan(k)) * Float64(2.0 / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-237)
		tmp = (l / sin(k)) * ((l / tan(k)) * (2.0 / (k * (k * t))));
	else
		tmp = 2.0 * (cos(k) / ((t * (sin(k) ^ 2.0)) / ((l / k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-237], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000002e-237

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. associate-*l* 28.9%

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

      3. associate-/r* 28.9%

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

      4. associate-/r/ 27.8%

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

      5. *-commutative 27.8%

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

      6. times-frac 30.0%

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

      7. +-commutative 30.0%

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

      8. associate--l+ 37.2%

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

      9. metadata-eval 37.2%

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

      10. +-rgt-identity 37.2%

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

      11. times-frac 49.1%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in t around 0 84.7%

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

      1. unpow2 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in k around 0 84.7%

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

      1. unpow2 84.7%

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

      2. associate-*r* 89.5%

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

   9. Simplified 89.5%

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

       1. expm1-log1p-u 82.4%

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

       2. expm1-udef 72.9%

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

       3. *-commutative 72.9%

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

       4. associate-/r* 72.9%

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

   11. Applied egg-rr 72.9%

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

       1. expm1-def 82.4%

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

       2. expm1-log1p 89.5%

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

       3. associate-*l* 94.5%

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

       4. associate-/l/ 94.5%

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

       5. *-commutative 94.5%

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

   13. Simplified 94.5%

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

   if 5.0000000000000002e-237 < (*.f64 l l)

   1. Initial program 38.6%

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

      1. associate-*l* 38.6%

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

      2. associate-*l* 38.6%

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

      3. associate-/r* 38.6%

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

      4. associate-/r/ 38.6%

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

      5. *-commutative 38.6%

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

      6. times-frac 38.1%

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

      7. +-commutative 38.1%

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

      8. associate--l+ 45.9%

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

      9. metadata-eval 45.9%

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

      10. +-rgt-identity 45.9%

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

      11. times-frac 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in t around 0 77.8%

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

      1. unpow2 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in k around 0 77.8%

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

      1. unpow2 77.8%

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

      2. associate-*r* 82.7%

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

   9. Simplified 82.7%

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

       1. expm1-log1p-u 47.4%

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

       2. expm1-udef 37.5%

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

       3. *-commutative 37.5%

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

       4. associate-/r* 37.6%

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

   11. Applied egg-rr 37.6%

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

       1. expm1-def 47.5%

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

       2. expm1-log1p 82.8%

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

       3. associate-*l* 87.6%

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

       4. associate-/l/ 87.4%

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

       5. *-commutative 87.4%

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

   13. Simplified 87.4%

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

   14. Taylor expanded in l around 0 78.3%

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

       1. associate-/l* 78.3%

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

       2. *-commutative 78.3%

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

       3. associate-/l* 79.6%

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

       4. unpow2 79.6%

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

       5. unpow2 79.6%

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

       6. times-frac 97.1%

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

       7. unpow2 97.1%

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

       8. *-commutative 97.1%

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

   16. Simplified 97.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 2: 77.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(t_1 \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* k (* k t)))))
   (if (<= k 3.1e-15)
     (* (/ l k) (* t_1 (/ l k)))
     (* t_1 (* (/ l (sin k)) (/ l (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / (k * (k * t));
	double tmp;
	if (k <= 3.1e-15) {
		tmp = (l / k) * (t_1 * (l / k));
	} else {
		tmp = t_1 * ((l / sin(k)) * (l / tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (k * (k * t))
    if (k <= 3.1d-15) then
        tmp = (l / k) * (t_1 * (l / k))
    else
        tmp = t_1 * ((l / sin(k)) * (l / tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / (k * (k * t));
	double tmp;
	if (k <= 3.1e-15) {
		tmp = (l / k) * (t_1 * (l / k));
	} else {
		tmp = t_1 * ((l / Math.sin(k)) * (l / Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / (k * (k * t))
	tmp = 0
	if k <= 3.1e-15:
		tmp = (l / k) * (t_1 * (l / k))
	else:
		tmp = t_1 * ((l / math.sin(k)) * (l / math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(k * Float64(k * t)))
	tmp = 0.0
	if (k <= 3.1e-15)
		tmp = Float64(Float64(l / k) * Float64(t_1 * Float64(l / k)));
	else
		tmp = Float64(t_1 * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / (k * (k * t));
	tmp = 0.0;
	if (k <= 3.1e-15)
		tmp = (l / k) * (t_1 * (l / k));
	else
		tmp = t_1 * ((l / sin(k)) * (l / tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.1e-15], N[(N[(l / k), $MachinePrecision] * N[(t$95$1 * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0999999999999999e-15

   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. associate-*l* 39.0%

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

      2. associate-*l* 39.0%

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

      3. associate-/r* 39.0%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. times-frac 39.1%

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

      7. +-commutative 39.1%

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

      8. associate--l+ 44.0%

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

      9. metadata-eval 44.0%

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

      10. +-rgt-identity 44.0%

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

      11. times-frac 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in t around 0 83.1%

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

      1. unpow2 83.1%

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

   6. Simplified 83.1%

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

   7. Taylor expanded in k around 0 68.3%

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

      1. unpow2 68.3%

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

      2. unpow2 68.3%

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

      3. times-frac 74.8%

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

   9. Simplified 74.8%

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

       1. associate-*r* 77.3%

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

       2. associate-*l/ 77.3%

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

       3. pow2 77.3%

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

   11. Applied egg-rr 77.3%

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

       1. pow2 77.3%

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

       2. associate-*l/ 77.3%

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

       3. associate-*r* 79.9%

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

   13. Applied egg-rr 79.9%

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

   if 3.0999999999999999e-15 < k

   1. Initial program 27.0%

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

      1. associate-*l* 27.0%

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

      2. associate-*l* 27.0%

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

      3. associate-/r* 27.0%

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

      4. associate-/r/ 27.0%

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

      5. *-commutative 27.0%

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

      6. times-frac 27.1%

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

      7. +-commutative 27.1%

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

      8. associate--l+ 40.5%

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

      9. metadata-eval 40.5%

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

      10. +-rgt-identity 40.5%

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

      11. times-frac 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in t around 0 73.9%

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

      1. unpow2 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in k around 0 73.9%

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

      1. unpow2 73.9%

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

      2. associate-*r* 80.1%

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

   9. Simplified 80.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \]

Alternative 3: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{if}\;k \leq 3 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(t_1 \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* k (* k t)))))
   (if (<= k 3e-28)
     (* (/ l k) (* t_1 (/ l k)))
     (* t_1 (/ (* l l) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / (k * (k * t));
	double tmp;
	if (k <= 3e-28) {
		tmp = (l / k) * (t_1 * (l / k));
	} else {
		tmp = t_1 * ((l * l) / (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (k * (k * t))
    if (k <= 3d-28) then
        tmp = (l / k) * (t_1 * (l / k))
    else
        tmp = t_1 * ((l * l) / (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / (k * (k * t));
	double tmp;
	if (k <= 3e-28) {
		tmp = (l / k) * (t_1 * (l / k));
	} else {
		tmp = t_1 * ((l * l) / (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / (k * (k * t))
	tmp = 0
	if k <= 3e-28:
		tmp = (l / k) * (t_1 * (l / k))
	else:
		tmp = t_1 * ((l * l) / (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(k * Float64(k * t)))
	tmp = 0.0
	if (k <= 3e-28)
		tmp = Float64(Float64(l / k) * Float64(t_1 * Float64(l / k)));
	else
		tmp = Float64(t_1 * Float64(Float64(l * l) / Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / (k * (k * t));
	tmp = 0.0;
	if (k <= 3e-28)
		tmp = (l / k) * (t_1 * (l / k));
	else
		tmp = t_1 * ((l * l) / (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3e-28], N[(N[(l / k), $MachinePrecision] * N[(t$95$1 * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.00000000000000003e-28

   1. Initial program 39.1%

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

      1. associate-*l* 39.1%

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

      2. associate-*l* 39.1%

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

      3. associate-/r* 39.1%

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

      4. associate-/r/ 38.5%

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

      5. *-commutative 38.5%

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

      6. times-frac 39.2%

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

      7. +-commutative 39.2%

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

      8. associate--l+ 44.2%

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

      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 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around 0 82.8%

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

      1. unpow2 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in k around 0 67.8%

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

      1. unpow2 67.8%

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

      2. unpow2 67.8%

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

      3. times-frac 74.4%

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

   9. Simplified 74.4%

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

       1. associate-*r* 76.9%

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

       2. associate-*l/ 76.9%

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

       3. pow2 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. pow2 76.9%

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

       2. associate-*l/ 76.9%

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

       3. associate-*r* 79.5%

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

   13. Applied egg-rr 79.5%

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

   if 3.00000000000000003e-28 < k

   1. Initial program 27.2%

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

      1. associate-*l* 27.3%

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

      2. associate-*l* 27.3%

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

      3. associate-/r* 27.3%

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

      4. associate-/r/ 27.3%

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

      5. *-commutative 27.3%

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

      6. times-frac 27.3%

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

      7. +-commutative 27.3%

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

      8. associate--l+ 40.2%

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

      9. metadata-eval 40.2%

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

      10. +-rgt-identity 40.2%

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

      11. times-frac 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around 0 74.8%

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

      1. unpow2 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in k around 0 74.8%

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

      1. unpow2 74.8%

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

      2. associate-*r* 80.8%

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

   9. Simplified 80.8%

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

       1. frac-times 80.9%

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

   11. Applied egg-rr 80.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}\\ \end{array} \]

Alternative 4: 88.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. associate-*l* 35.2%

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

   2. associate-*l* 35.2%

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

   3. associate-/r* 35.2%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.3%

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

   7. +-commutative 35.3%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 80.2%

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

   1. unpow2 80.2%

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

6. Simplified 80.2%

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

7. Taylor expanded in k around 0 80.2%

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

   1. unpow2 80.2%

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

   2. associate-*r* 85.1%

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

9. Simplified 85.1%

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

    1. expm1-log1p-u 59.7%

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

    2. expm1-udef 50.0%

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

    3. *-commutative 50.0%

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

    4. associate-/r* 50.0%

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

11. Applied egg-rr 50.0%

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

    1. expm1-def 59.8%

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

    2. expm1-log1p 85.1%

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

    3. associate-*l* 90.0%

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

    4. associate-/l/ 89.9%

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

    5. *-commutative 89.9%

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

13. Simplified 89.9%

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

14. Final simplification 89.9%

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

Alternative 5: 89.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. associate-*l* 35.2%

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

   2. associate-*l* 35.2%

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

   3. associate-/r* 35.2%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.3%

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

   7. +-commutative 35.3%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 80.2%

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

   1. unpow2 80.2%

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

6. Simplified 80.2%

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

7. Taylor expanded in k around 0 80.2%

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

   1. unpow2 80.2%

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

   2. associate-*r* 85.1%

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

9. Simplified 85.1%

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

    1. expm1-log1p-u 59.7%

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

    2. expm1-udef 50.0%

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

    3. *-commutative 50.0%

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

    4. associate-/r* 50.0%

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

11. Applied egg-rr 50.0%

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

    1. expm1-def 59.8%

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

    2. expm1-log1p 85.1%

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

    3. associate-*l* 90.0%

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

    4. associate-/l/ 89.9%

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

    5. *-commutative 89.9%

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

13. Simplified 89.9%

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

    1. frac-times 87.7%

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

15. Applied egg-rr 87.7%

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

    1. times-frac 89.9%

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

    2. associate-/r* 90.0%

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

17. Simplified 90.0%

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

18. Final simplification 90.0%

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

Alternative 6: 70.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 350:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{2}{t_1}\right) \cdot \left(\frac{\ell}{k} + 0.16666666666666666 \cdot \left(\ell \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{t_1} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 350.0)
     (*
      (* (/ l (tan k)) (/ 2.0 t_1))
      (+ (/ l k) (* 0.16666666666666666 (* l k))))
     (*
      2.0
      (*
       (/ (cos k) (* k k))
       (+ (/ (* l l) t_1) (* 0.3333333333333333 (/ (* l l) t))))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 350.0) {
		tmp = ((l / tan(k)) * (2.0 / t_1)) * ((l / k) + (0.16666666666666666 * (l * k)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (((l * l) / t_1) + (0.3333333333333333 * ((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) :: t_1
    real(8) :: tmp
    t_1 = k * (k * t)
    if (k <= 350.0d0) then
        tmp = ((l / tan(k)) * (2.0d0 / t_1)) * ((l / k) + (0.16666666666666666d0 * (l * k)))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (((l * l) / t_1) + (0.3333333333333333d0 * ((l * l) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 350.0) {
		tmp = ((l / Math.tan(k)) * (2.0 / t_1)) * ((l / k) + (0.16666666666666666 * (l * k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (((l * l) / t_1) + (0.3333333333333333 * ((l * l) / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 350.0:
		tmp = ((l / math.tan(k)) * (2.0 / t_1)) * ((l / k) + (0.16666666666666666 * (l * k)))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (((l * l) / t_1) + (0.3333333333333333 * ((l * l) / t))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 350.0)
		tmp = Float64(Float64(Float64(l / tan(k)) * Float64(2.0 / t_1)) * Float64(Float64(l / k) + Float64(0.16666666666666666 * Float64(l * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(Float64(l * l) / t_1) + Float64(0.3333333333333333 * Float64(Float64(l * l) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 350.0)
		tmp = ((l / tan(k)) * (2.0 / t_1)) * ((l / k) + (0.16666666666666666 * (l * k)));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * (((l * l) / t_1) + (0.3333333333333333 * ((l * l) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 350.0], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] + N[(0.16666666666666666 * N[(l * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 350

   1. Initial program 38.5%

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

      1. associate-*l* 38.5%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.5%

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

      4. associate-/r/ 38.0%

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

      5. *-commutative 38.0%

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

      6. times-frac 38.7%

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

      7. +-commutative 38.7%

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

      8. associate--l+ 44.0%

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

      9. metadata-eval 44.0%

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

      10. +-rgt-identity 44.0%

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

      11. times-frac 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in t around 0 83.3%

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

      1. unpow2 83.3%

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

   6. Simplified 83.3%

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

   7. Taylor expanded in k around 0 83.3%

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

      1. unpow2 83.3%

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

      2. associate-*r* 87.5%

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

   9. Simplified 87.5%

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

       1. expm1-log1p-u 54.9%

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

       2. expm1-udef 45.6%

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

       3. *-commutative 45.6%

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

       4. associate-/r* 45.6%

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

   11. Applied egg-rr 45.6%

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

       1. expm1-def 55.0%

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

       2. expm1-log1p 87.6%

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

       3. associate-*l* 91.4%

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

       4. associate-/l/ 91.1%

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

       5. *-commutative 91.1%

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

   13. Simplified 91.1%

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

   14. Taylor expanded in k around 0 76.6%

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

   if 350 < k

   1. Initial program 27.7%

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

      1. associate-*l* 27.7%

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

      2. associate-*l* 27.7%

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

      3. associate-/r* 27.7%

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

      4. associate-/r/ 27.7%

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

      5. *-commutative 27.7%

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

      6. times-frac 27.7%

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

      7. +-commutative 27.7%

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

      8. associate--l+ 40.2%

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

      9. metadata-eval 40.2%

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

      10. +-rgt-identity 40.2%

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

      11. times-frac 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around 0 73.3%

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

      1. unpow2 73.3%

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

      2. times-frac 72.1%

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

      3. unpow2 72.1%

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

      4. *-commutative 72.1%

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

      5. associate-/r* 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in k around 0 55.3%

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

      1. unpow2 55.3%

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

      2. unpow2 55.3%

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

      3. associate-*r* 55.3%

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

      4. unpow2 55.3%

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

   9. Simplified 55.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 350:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right) \cdot \left(\frac{\ell}{k} + 0.16666666666666666 \cdot \left(\ell \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \end{array} \]

Alternative 7: 72.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{1}{\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e-108)
   (* (/ l k) (* (/ 2.0 (* k (* k t))) (/ l k)))
   (* 2.0 (* (/ (cos k) (* k k)) (/ 1.0 (* (/ t l) (/ (* k k) l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-108) {
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (1.0 / ((t / l) * ((k * k) / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d-108) then
        tmp = (l / k) * ((2.0d0 / (k * (k * t))) * (l / k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (1.0d0 / ((t / l) * ((k * k) / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-108) {
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (1.0 / ((t / l) * ((k * k) / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e-108:
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (1.0 / ((t / l) * ((k * k) / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e-108)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(1.0 / Float64(Float64(t / l) * Float64(Float64(k * k) / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e-108)
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * (1.0 / ((t / l) * ((k * k) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e-108], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-108

   1. Initial program 41.3%

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

      1. associate-*l* 41.3%

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

      2. associate-*l* 41.3%

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

      3. associate-/r* 41.4%

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

      4. associate-/r/ 40.7%

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

      5. *-commutative 40.7%

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

      6. times-frac 41.5%

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

      7. +-commutative 41.5%

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

      8. associate--l+ 46.7%

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

      9. metadata-eval 46.7%

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

      10. +-rgt-identity 46.7%

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

      11. times-frac 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in t around 0 82.7%

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

      1. unpow2 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in k around 0 66.4%

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

      1. unpow2 66.4%

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

      2. unpow2 66.4%

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

      3. times-frac 73.4%

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

   9. Simplified 73.4%

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

       1. associate-*r* 76.2%

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

       2. associate-*l/ 76.2%

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

       3. pow2 76.2%

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

   11. Applied egg-rr 76.2%

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

       1. pow2 76.2%

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

       2. associate-*l/ 76.2%

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

       3. associate-*r* 77.9%

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

   13. Applied egg-rr 77.9%

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

   if 1.45e-108 < k

   1. Initial program 25.7%

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

      1. associate-*l* 25.8%

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

      2. associate-*l* 25.8%

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

      3. associate-/r* 25.8%

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

      4. associate-/r/ 25.8%

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

      5. *-commutative 25.8%

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

      6. times-frac 25.8%

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

      7. +-commutative 25.8%

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

      8. associate--l+ 36.9%

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

      9. metadata-eval 36.9%

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

      10. +-rgt-identity 36.9%

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

      11. times-frac 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in t around 0 76.6%

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

      1. unpow2 76.6%

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

      2. times-frac 75.4%

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

      3. unpow2 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-/r* 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in k around 0 60.8%

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

      1. unpow2 60.8%

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

      2. unpow2 60.8%

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

      3. associate-*r* 60.8%

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

   9. Simplified 60.8%

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

       1. clear-num 60.2%

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

       2. inv-pow 60.2%

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

   11. Applied egg-rr 60.2%

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

       1. unpow-1 60.2%

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

       2. associate-*r* 60.2%

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

       3. unpow2 60.2%

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

       4. *-commutative 60.2%

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

       5. times-frac 62.9%

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

       6. unpow2 62.9%

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

   13. Simplified 62.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{1}{\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}}\right)\\ \end{array} \]

Alternative 8: 72.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.5e-251)
   (* (/ l k) (* (/ 2.0 (* k (* k t))) (/ l k)))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l k) (/ l (* k t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.5e-251) {
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / k) * (l / (k * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 2.5d-251) then
        tmp = (l / k) * ((2.0d0 / (k * (k * t))) * (l / k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / k) * (l / (k * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.5e-251) {
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / k) * (l / (k * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 2.5e-251:
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / k) * (l / (k * t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.5e-251)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / k) * Float64(l / Float64(k * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.5e-251)
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / k) * (l / (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.5e-251], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.5000000000000001e-251

   1. Initial program 35.8%

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

      1. associate-*l* 35.8%

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

      2. associate-*l* 35.8%

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

      3. associate-/r* 35.8%

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

      4. associate-/r/ 35.8%

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

      5. *-commutative 35.8%

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

      6. times-frac 36.5%

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

      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+ 45.0%

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

      9. metadata-eval 45.0%

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

      10. +-rgt-identity 45.0%

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

      11. times-frac 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 81.4%

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

      1. unpow2 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in k around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. times-frac 73.2%

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

   9. Simplified 73.2%

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

       1. associate-*r* 76.2%

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

       2. associate-*l/ 76.2%

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

       3. pow2 76.2%

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

   11. Applied egg-rr 76.2%

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

       1. pow2 76.2%

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

       2. associate-*l/ 76.2%

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

       3. associate-*r* 78.7%

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

   13. Applied egg-rr 78.7%

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

   if 2.5000000000000001e-251 < l

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-*l* 34.4%

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

      3. associate-/r* 34.4%

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

      4. associate-/r/ 33.5%

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

      5. *-commutative 33.5%

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

      6. times-frac 33.7%

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

      7. +-commutative 33.7%

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

      8. associate--l+ 40.1%

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

      9. metadata-eval 40.1%

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

      10. +-rgt-identity 40.1%

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

      11. times-frac 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around 0 79.4%

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

      1. unpow2 79.4%

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

      2. times-frac 80.2%

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

      3. unpow2 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-/r* 79.5%

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

   6. Simplified 79.5%

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

   7. Taylor expanded in k around 0 62.6%

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

      1. unpow2 62.6%

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

      2. unpow2 62.6%

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

      3. associate-*r* 62.6%

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

   9. Simplified 62.6%

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

       1. times-frac 63.9%

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

   11. Applied egg-rr 63.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-251}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\right)\\ \end{array} \]

Alternative 9: 72.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.5e-109)
   (* (/ l k) (* (/ 2.0 (* k (* k t))) (/ l k)))
   (* 2.0 (* (/ (cos k) (* k k)) (* (/ l t) (/ l (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.5e-109) {
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.5d-109) then
        tmp = (l / k) * ((2.0d0 / (k * (k * t))) * (l / k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.5e-109) {
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.5e-109:
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.5e-109)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.5e-109)
		tmp = (l / k) * ((2.0 / (k * (k * t))) * (l / k));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.5e-109], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.49999999999999959e-109

   1. Initial program 41.3%

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

      1. associate-*l* 41.3%

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

      2. associate-*l* 41.3%

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

      3. associate-/r* 41.4%

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

      4. associate-/r/ 40.7%

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

      5. *-commutative 40.7%

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

      6. times-frac 41.5%

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

      7. +-commutative 41.5%

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

      8. associate--l+ 46.7%

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

      9. metadata-eval 46.7%

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

      10. +-rgt-identity 46.7%

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

      11. times-frac 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in t around 0 82.7%

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

      1. unpow2 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in k around 0 66.4%

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

      1. unpow2 66.4%

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

      2. unpow2 66.4%

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

      3. times-frac 73.4%

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

   9. Simplified 73.4%

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

       1. associate-*r* 76.2%

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

       2. associate-*l/ 76.2%

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

       3. pow2 76.2%

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

   11. Applied egg-rr 76.2%

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

       1. pow2 76.2%

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

       2. associate-*l/ 76.2%

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

       3. associate-*r* 77.9%

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

   13. Applied egg-rr 77.9%

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

   if 6.49999999999999959e-109 < k

   1. Initial program 25.7%

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

      1. associate-*l* 25.8%

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

      2. associate-*l* 25.8%

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

      3. associate-/r* 25.8%

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

      4. associate-/r/ 25.8%

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

      5. *-commutative 25.8%

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

      6. times-frac 25.8%

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

      7. +-commutative 25.8%

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

      8. associate--l+ 36.9%

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

      9. metadata-eval 36.9%

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

      10. +-rgt-identity 36.9%

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

      11. times-frac 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in t around 0 76.6%

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

      1. unpow2 76.6%

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

      2. times-frac 75.4%

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

      3. unpow2 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-/r* 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in k around 0 60.8%

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

      1. unpow2 60.8%

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

      2. *-commutative 60.8%

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

      3. times-frac 62.8%

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

      4. unpow2 62.8%

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

   9. Simplified 62.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 10: 70.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

   1. associate-*l* 35.2%

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

   2. associate-*l* 35.2%

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

   3. associate-/r* 35.2%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.3%

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

   7. +-commutative 35.3%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 80.2%

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

   1. unpow2 80.2%

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

6. Simplified 80.2%

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

7. Taylor expanded in k around 0 63.7%

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

   1. unpow2 63.7%

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

   2. unpow2 63.7%

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

   3. times-frac 68.0%

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

9. Simplified 68.0%

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

    1. associate-*r* 69.7%

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

    2. associate-*l/ 69.6%

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

    3. pow2 69.6%

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

11. Applied egg-rr 69.6%

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

    1. pow2 69.6%

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

    2. associate-*l/ 69.7%

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

    3. associate-*r* 71.4%

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

13. Applied egg-rr 71.4%

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

14. Final simplification 71.4%

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

Alternative 11: 19.6% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ -0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.11666666666666667 (/ l (/ t l))))

C

double code(double t, double l, double k) {
	return -0.11666666666666667 * (l / (t / 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 = (-0.11666666666666667d0) * (l / (t / l))
end function

Java

public static double code(double t, double l, double k) {
	return -0.11666666666666667 * (l / (t / l));
}

Python

def code(t, l, k):
	return -0.11666666666666667 * (l / (t / l))

Julia

function code(t, l, k)
	return Float64(-0.11666666666666667 * Float64(l / Float64(t / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.11666666666666667 * (l / (t / l));
end

Wolfram

code[t_, l_, k_] := N[(-0.11666666666666667 * N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

   1. associate-*l* 35.2%

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

   2. associate-*l* 35.2%

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

   3. associate-/r* 35.2%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.3%

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

   7. +-commutative 35.3%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 80.2%

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

   1. unpow2 80.2%

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

6. Simplified 80.2%

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

7. Taylor expanded in k around 0 31.6%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \left(\frac{{\ell}^{2}}{{k}^{2}} + -1 \cdot \left({k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-+r+ 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) + -1 \cdot \left({k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)\right)} \]

   2. mul-1-neg 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) + \color{blue}{\left(-{k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)}\right) \]

   3. unsub-neg 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)} \]

   4. *-commutative 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   5. fma-def 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   6. unpow2 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   7. unpow2 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   8. unpow2 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   9. times-frac 36.4%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   10. unpow2 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \color{blue}{\left(k \cdot k\right)} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   11. distribute-rgt-out 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \left(0.08611111111111111 + -0.027777777777777776\right)\right)}\right) \]

   12. unpow2 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.08611111111111111 + -0.027777777777777776\right)\right)\right) \]

   13. metadata-eval 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{0.058333333333333334}\right)\right) \]

9. Simplified 36.4%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.058333333333333334\right)\right)} \]

10. Taylor expanded in k around inf 23.4%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{{\ell}^{2}}{t}} \]
11. Step-by-step derivation

    1. unpow2 23.4%

       \[\leadsto -0.11666666666666667 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

    2. associate-/l* 20.5%

       \[\leadsto -0.11666666666666667 \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}} \]

12. Simplified 20.5%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}} \]

13. Final simplification 20.5%

    \[\leadsto -0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}} \]

Alternative 12: 22.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (* l l) t) -0.11666666666666667))

C

double code(double t, double l, double k) {
	return ((l * l) / t) * -0.11666666666666667;
}

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 * l) / t) * (-0.11666666666666667d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((l * l) / t) * -0.11666666666666667;
}

Python

def code(t, l, k):
	return ((l * l) / t) * -0.11666666666666667

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * l) / t) * -0.11666666666666667)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * l) / t) * -0.11666666666666667;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.11666666666666667), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

   1. associate-*l* 35.2%

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

   2. associate-*l* 35.2%

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

   3. associate-/r* 35.2%

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

   4. associate-/r/ 34.8%

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

   5. *-commutative 34.8%

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

   6. times-frac 35.3%

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

   7. +-commutative 35.3%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 80.2%

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

   1. unpow2 80.2%

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

6. Simplified 80.2%

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

7. Taylor expanded in k around 0 31.6%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \left(\frac{{\ell}^{2}}{{k}^{2}} + -1 \cdot \left({k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-+r+ 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) + -1 \cdot \left({k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)\right)} \]

   2. mul-1-neg 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) + \color{blue}{\left(-{k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)}\right) \]

   3. unsub-neg 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right)} \]

   4. *-commutative 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   5. fma-def 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   6. unpow2 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   7. unpow2 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   8. unpow2 31.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   9. times-frac 36.4%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) - {k}^{2} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   10. unpow2 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \color{blue}{\left(k \cdot k\right)} \cdot \left(0.08611111111111111 \cdot {\ell}^{2} + -0.027777777777777776 \cdot {\ell}^{2}\right)\right) \]

   11. distribute-rgt-out 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \color{blue}{\left({\ell}^{2} \cdot \left(0.08611111111111111 + -0.027777777777777776\right)\right)}\right) \]

   12. unpow2 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.08611111111111111 + -0.027777777777777776\right)\right)\right) \]

   13. metadata-eval 36.4%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{0.058333333333333334}\right)\right) \]

9. Simplified 36.4%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right) - \left(k \cdot k\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.058333333333333334\right)\right)} \]

10. Taylor expanded in k around inf 23.4%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{{\ell}^{2}}{t}} \]
11. Step-by-step derivation

    1. unpow2 23.4%

       \[\leadsto -0.11666666666666667 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

12. Simplified 23.4%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t}} \]

13. Final simplification 23.4%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667 \]

Reproduce

herbie shell --seed 2023192 
(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))))