Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 86.1%

Time: 14.7s

Alternatives: 6

Speedup: 32.4×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

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: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.1e-7)
   (* (/ (/ l t) (* k t)) (/ (/ l k) t))
   (* 2.0 (/ (* (pow (/ l k) 2.0) (cos k)) (* t (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e-7) {
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) * cos(k)) / (t * pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.1d-7) then
        tmp = ((l / t) / (k * t)) * ((l / k) / t)
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) * cos(k)) / (t * (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e-7) {
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * Math.cos(k)) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.1e-7:
		tmp = ((l / t) / (k * t)) * ((l / k) / t)
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) * math.cos(k)) / (t * math.pow(math.sin(k), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.1e-7)
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * Float64(Float64(l / k) / t));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * cos(k)) / Float64(t * (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.1e-7)
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) * cos(k)) / (t * (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.1e-7], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e-7

   1. Initial program 60.7%

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

      1. associate-*l* 61.2%

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

      2. associate-/l/ 61.2%

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

      3. *-commutative 61.2%

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

      4. associate-*r/ 62.0%

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

      5. associate-/l* 61.2%

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

      6. associate-/r/ 52.1%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in k around 0 54.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 53.9%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 72.3%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. pow2 72.3%

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

      2. pow-to-exp 36.1%

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

   8. Applied egg-rr 36.1%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
   9. Step-by-step derivation

      1. pow-to-exp 72.3%

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

      2. unpow2 72.3%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

      3. unpow3 72.3%

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

      4. times-frac 76.9%

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

   10. Applied egg-rr 76.9%

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

   11. Taylor expanded in l around 0 76.4%

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

       1. unpow2 76.4%

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

       2. *-commutative 76.4%

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

       3. associate-*l* 80.9%

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

       4. associate-/r* 81.2%

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

       5. *-commutative 81.2%

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

   13. Simplified 81.2%

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

   if 1.1000000000000001e-7 < k

   1. Initial program 48.1%

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

      1. associate-*l* 48.1%

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

      2. associate-/l/ 48.1%

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

      3. *-commutative 48.1%

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

      4. associate-*r/ 48.1%

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

      5. associate-/l* 48.1%

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

      6. associate-/r/ 48.1%

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

   3. Simplified 51.3%

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

      1. unpow2 51.3%

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

   5. Applied egg-rr 51.3%

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

   6. Taylor expanded in k around inf 66.9%

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

      1. *-commutative 66.9%

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

      2. times-frac 65.6%

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

      3. unpow2 65.6%

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

      4. unpow2 65.6%

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

      5. times-frac 83.1%

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

      6. unpow2 83.1%

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

      7. *-commutative 83.1%

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

   8. Simplified 83.1%

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

      1. associate-*r/ 83.1%

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

   10. Applied egg-rr 83.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 2: 86.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.55e-6)
   (* (/ (/ l t) (* k t)) (/ (/ l k) t))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-6) {
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.55d-6) then
        tmp = ((l / t) / (k * t)) * ((l / k) / t)
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-6) {
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.55e-6:
		tmp = ((l / t) / (k * t)) * ((l / k) / t)
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.55e-6)
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * Float64(Float64(l / k) / t));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.55e-6)
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.55e-6], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55e-6

   1. Initial program 60.7%

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

      1. associate-*l* 61.2%

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

      2. associate-/l/ 61.2%

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

      3. *-commutative 61.2%

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

      4. associate-*r/ 62.0%

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

      5. associate-/l* 61.2%

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

      6. associate-/r/ 52.1%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in k around 0 54.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.9%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 53.9%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 72.3%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. pow2 72.3%

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

      2. pow-to-exp 36.1%

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

   8. Applied egg-rr 36.1%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
   9. Step-by-step derivation

      1. pow-to-exp 72.3%

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

      2. unpow2 72.3%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

      3. unpow3 72.3%

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

      4. times-frac 76.9%

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

   10. Applied egg-rr 76.9%

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

   11. Taylor expanded in l around 0 76.4%

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

       1. unpow2 76.4%

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

       2. *-commutative 76.4%

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

       3. associate-*l* 80.9%

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

       4. associate-/r* 81.2%

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

       5. *-commutative 81.2%

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

   13. Simplified 81.2%

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

   if 1.55e-6 < k

   1. Initial program 48.1%

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

      1. associate-*l* 48.1%

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

      2. associate-/l/ 48.1%

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

      3. *-commutative 48.1%

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

      4. associate-*r/ 48.1%

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

      5. associate-/l* 48.1%

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

      6. associate-/r/ 48.1%

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

   3. Simplified 51.3%

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

      1. unpow2 51.3%

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

   5. Applied egg-rr 51.3%

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

   6. Taylor expanded in k around inf 66.9%

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

      1. *-commutative 66.9%

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

      2. times-frac 65.6%

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

      3. unpow2 65.6%

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

      4. unpow2 65.6%

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

      5. times-frac 83.1%

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

      6. unpow2 83.1%

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

      7. *-commutative 83.1%

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

   8. Simplified 83.1%

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

      1. unpow2 67.4%

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

   10. Applied egg-rr 83.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-40}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ l k) t)))
   (if (<= t -3.8e-40)
     (* t_1 (/ l (* t (* k t))))
     (if (<= t 1.35e-57)
       (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (* (sin k) (tan k))))
       (* (/ (/ l t) (* k t)) t_1)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (l / k) / t;
	double tmp;
	if (t <= -3.8e-40) {
		tmp = t_1 * (l / (t * (k * t)));
	} else if (t <= 1.35e-57) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (sin(k) * tan(k)));
	} else {
		tmp = ((l / t) / (k * t)) * t_1;
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) / t
    if (t <= (-3.8d-40)) then
        tmp = t_1 * (l / (t * (k * t)))
    else if (t <= 1.35d-57) then
        tmp = 2.0d0 / (((t / l) * ((k * k) / l)) * (sin(k) * tan(k)))
    else
        tmp = ((l / t) / (k * t)) * t_1
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (l / k) / t;
	double tmp;
	if (t <= -3.8e-40) {
		tmp = t_1 * (l / (t * (k * t)));
	} else if (t <= 1.35e-57) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = ((l / t) / (k * t)) * t_1;
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (l / k) / t
	tmp = 0
	if t <= -3.8e-40:
		tmp = t_1 * (l / (t * (k * t)))
	elif t <= 1.35e-57:
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (math.sin(k) * math.tan(k)))
	else:
		tmp = ((l / t) / (k * t)) * t_1
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(l / k) / t)
	tmp = 0.0
	if (t <= -3.8e-40)
		tmp = Float64(t_1 * Float64(l / Float64(t * Float64(k * t))));
	elseif (t <= 1.35e-57)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * t_1);
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) / t;
	tmp = 0.0;
	if (t <= -3.8e-40)
		tmp = t_1 * (l / (t * (k * t)));
	elseif (t <= 1.35e-57)
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (sin(k) * tan(k)));
	else
		tmp = ((l / t) / (k * t)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -3.8e-40], N[(t$95$1 * N[(l / N[(t * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-57], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.7999999999999999e-40

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. associate-/l/ 60.5%

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

      3. *-commutative 60.5%

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

      4. associate-*r/ 63.9%

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

      5. associate-/l* 60.5%

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

      6. associate-/r/ 48.9%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in k around 0 52.3%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 52.2%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 52.2%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 52.2%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 73.7%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. pow2 73.7%

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

      2. pow-to-exp 41.0%

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

   8. Applied egg-rr 41.0%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
   9. Step-by-step derivation

      1. pow-to-exp 73.7%

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

      2. unpow2 73.7%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

      3. unpow3 73.8%

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

      4. times-frac 78.7%

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

   10. Applied egg-rr 78.7%

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

   11. Taylor expanded in l around 0 78.7%

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

       1. unpow2 78.7%

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

       2. *-commutative 78.7%

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

       3. associate-*l* 80.0%

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

       4. *-commutative 80.0%

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

   13. Simplified 80.0%

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

   if -3.7999999999999999e-40 < t < 1.3500000000000001e-57

   1. Initial program 43.1%

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

      1. *-commutative 43.1%

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

      2. associate-*l* 43.0%

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

      3. associate-*r* 44.0%

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

      4. +-commutative 44.0%

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

      5. associate-+r+ 44.0%

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

      6. metadata-eval 44.0%

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

   3. Simplified 44.0%

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

   4. Taylor expanded in k around inf 73.1%

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

      1. *-commutative 73.1%

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

      2. unpow2 73.1%

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

      3. times-frac 86.1%

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

      4. unpow2 86.1%

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

   6. Simplified 86.1%

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

   if 1.3500000000000001e-57 < t

   1. Initial program 73.4%

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

      1. associate-*l* 73.3%

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

      2. associate-/l/ 73.3%

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

      3. *-commutative 73.3%

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

      4. associate-*r/ 73.3%

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

      5. associate-/l* 73.4%

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

      6. associate-/r/ 62.8%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 60.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 59.6%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 59.6%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 59.6%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 70.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. pow2 70.6%

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

      2. pow-to-exp 37.0%

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

   8. Applied egg-rr 37.0%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
   9. Step-by-step derivation

      1. pow-to-exp 70.6%

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

      2. unpow2 70.6%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

      3. unpow3 70.6%

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

      4. times-frac 75.7%

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

   10. Applied egg-rr 75.7%

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

   11. Taylor expanded in l around 0 74.5%

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

       1. unpow2 74.5%

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

       2. *-commutative 74.5%

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

       3. associate-*l* 76.9%

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

       4. associate-/r* 78.1%

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

       5. *-commutative 78.1%

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

   13. Simplified 78.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{t \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 4: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.4e+52)
   (* (/ (/ l t) (* k t)) (/ (/ l k) t))
   (pow (/ (cbrt (* (/ l k) (/ l k))) t) 3.0)))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e+52) {
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	} else {
		tmp = pow((cbrt(((l / k) * (l / k))) / t), 3.0);
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.4e+52) {
		tmp = ((l / t) / (k * t)) * ((l / k) / t);
	} else {
		tmp = Math.pow((Math.cbrt(((l / k) * (l / k))) / t), 3.0);
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.4e+52)
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * Float64(Float64(l / k) / t));
	else
		tmp = Float64(cbrt(Float64(Float64(l / k) * Float64(l / k))) / t) ^ 3.0;
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.4e+52], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e52

   1. Initial program 59.9%

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

      1. associate-*l* 60.4%

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

      2. associate-/l/ 60.4%

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

      3. *-commutative 60.4%

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

      4. associate-*r/ 61.1%

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

      5. associate-/l* 60.4%

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

      6. associate-/r/ 51.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in k around 0 53.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 53.6%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 53.6%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 53.6%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 70.9%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. pow2 70.9%

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

      2. pow-to-exp 35.2%

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

   8. Applied egg-rr 35.2%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
   9. Step-by-step derivation

      1. pow-to-exp 70.9%

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

      2. unpow2 70.9%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

      3. unpow3 70.9%

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

      4. times-frac 75.3%

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

   10. Applied egg-rr 75.3%

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

   11. Taylor expanded in l around 0 74.8%

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

       1. unpow2 74.8%

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

       2. *-commutative 74.8%

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

       3. associate-*l* 79.1%

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

       4. associate-/r* 79.9%

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

       5. *-commutative 79.9%

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

   13. Simplified 79.9%

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

   if 2.4e52 < k

   1. Initial program 48.5%

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

      1. associate-*l* 48.5%

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

      2. associate-/l/ 48.5%

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

      3. *-commutative 48.5%

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

      4. associate-*r/ 48.5%

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

      5. associate-/l* 48.5%

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

      6. associate-/r/ 48.5%

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

   3. Simplified 50.9%

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

   4. Taylor expanded in k around 0 46.7%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. associate-/r* 48.6%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

      3. unpow2 48.6%

         \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

      4. times-frac 53.2%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   6. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   7. Step-by-step derivation

      1. pow2 53.2%

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

      2. pow-to-exp 40.8%

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

   8. Applied egg-rr 40.8%

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
   9. Step-by-step derivation

      1. add-cube-cbrt 40.8%

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

      2. pow3 40.8%

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

      3. pow-to-exp 53.2%

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

      4. cbrt-div 53.2%

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

      5. rem-cbrt-cube 69.7%

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

   10. Applied egg-rr 69.7%

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{{\left(\frac{\ell}{k}\right)}^{2}}}{t}\right)}^{3}} \]
   11. Step-by-step derivation

       1. unpow2 69.7%

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

   12. Applied egg-rr 69.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\frac{\ell}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}\right)}^{3}\\ \end{array} \]

Alternative 5: 69.4% accurate, 32.4× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* (/ (/ l k) t) (/ l (* t (* k t)))))

C

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

Fortran

NOTE: k should be positive before calling this function
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) / t) * (l / (t * (k * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(t * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 58.1%

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

   2. associate-/l/ 58.1%

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

   3. *-commutative 58.1%

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

   4. associate-*r/ 58.7%

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

   5. associate-/l* 58.1%

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

   6. associate-/r/ 51.2%

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

3. Simplified 58.1%

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

4. Taylor expanded in k around 0 52.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. associate-/r* 52.6%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

   2. unpow2 52.6%

      \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

   3. unpow2 52.6%

      \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

   4. times-frac 67.5%

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

6. Simplified 67.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
7. Step-by-step derivation

   1. pow2 67.5%

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

   2. pow-to-exp 36.3%

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

8. Applied egg-rr 36.3%

   \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
9. Step-by-step derivation

   1. pow-to-exp 67.5%

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

   2. unpow2 67.5%

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   3. unpow3 67.5%

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

   4. times-frac 71.0%

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

10. Applied egg-rr 71.0%

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

11. Taylor expanded in l around 0 70.6%

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

    1. unpow2 70.6%

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

    2. *-commutative 70.6%

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

    3. associate-*l* 75.7%

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

    4. *-commutative 75.7%

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

13. Simplified 75.7%

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

14. Final simplification 75.7%

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

Alternative 6: 71.2% accurate, 32.4× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* (/ (/ l t) (* k t)) (/ (/ l k) t)))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / t) / (k * t)) * ((l / k) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 58.1%

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

   2. associate-/l/ 58.1%

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

   3. *-commutative 58.1%

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

   4. associate-*r/ 58.7%

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

   5. associate-/l* 58.1%

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

   6. associate-/r/ 51.2%

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

3. Simplified 58.1%

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

4. Taylor expanded in k around 0 52.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
5. Step-by-step derivation

   1. associate-/r* 52.6%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{{t}^{3}}} \]

   2. unpow2 52.6%

      \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{t}^{3}} \]

   3. unpow2 52.6%

      \[\leadsto \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{t}^{3}} \]

   4. times-frac 67.5%

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

6. Simplified 67.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
7. Step-by-step derivation

   1. pow2 67.5%

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

   2. pow-to-exp 36.3%

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

8. Applied egg-rr 36.3%

   \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\ell}{k}\right) \cdot 2}}}{{t}^{3}} \]
9. Step-by-step derivation

   1. pow-to-exp 67.5%

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

   2. unpow2 67.5%

      \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{{t}^{3}} \]

   3. unpow3 67.5%

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

   4. times-frac 71.0%

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

10. Applied egg-rr 71.0%

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

11. Taylor expanded in l around 0 70.6%

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

    1. unpow2 70.6%

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

    2. *-commutative 70.6%

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

    3. associate-*l* 75.7%

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

    4. associate-/r* 77.1%

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

    5. *-commutative 77.1%

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

13. Simplified 77.1%

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

14. Final simplification 77.1%

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

Reproduce

herbie shell --seed 2023257 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))