Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.1% → 96.5%

Time: 31.6s

Alternatives: 15

Speedup: 24.7×

• 33.8% 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: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. associate-/r* 30.8%

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

   2. *-commutative 30.8%

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

   3. associate-/r* 35.0%

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

   4. associate-*r/ 35.8%

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

   5. associate-/l* 35.0%

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

   6. +-commutative 35.0%

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

   7. unpow2 35.0%

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

   8. sqr-neg 35.0%

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

   9. distribute-frac-neg 35.0%

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

   10. distribute-frac-neg 35.0%

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

   11. unpow2 35.0%

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

   12. associate--l+ 45.2%

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

   13. metadata-eval 45.2%

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

   14. +-rgt-identity 45.2%

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

   15. unpow2 45.2%

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

   16. distribute-frac-neg 45.2%

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

3. Simplified 45.2%

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

4. Taylor expanded in k around inf 65.6%

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

   1. associate-*r* 65.6%

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

   2. associate-/r* 67.2%

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

   3. unpow2 67.2%

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

   4. associate-*l* 67.2%

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

   5. unpow2 67.2%

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

   6. associate-*l* 71.1%

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

6. Simplified 71.1%

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

7. Taylor expanded in l around 0 67.2%

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

   1. associate-/l* 67.2%

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

   2. associate-/r/ 67.2%

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

   3. associate-/r* 67.0%

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

   4. unpow2 67.0%

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

   5. unpow2 67.0%

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

   6. times-frac 88.5%

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

   7. *-lft-identity 88.5%

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

   8. associate-*l/ 88.5%

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

   9. *-commutative 88.5%

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

   10. associate-*l* 95.5%

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

   11. associate-*r/ 95.6%

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

   12. *-commutative 95.6%

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

   13. *-lft-identity 95.6%

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

9. Simplified 95.6%

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

10. Final simplification 95.6%

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

Alternative 2: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.85e-5)
   (* 2.0 (/ (* (* (/ l k) (/ (/ l k) t)) (cos k)) (* k k)))
   (* 2.0 (/ (* (/ (cos k) t) (* l (/ (/ l k) k))) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-5) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	} else {
		tmp = 2.0 * (((cos(k) / t) * (l * ((l / k) / k))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.85d-5) then
        tmp = 2.0d0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k))
    else
        tmp = 2.0d0 * (((cos(k) / t) * (l * ((l / k) / k))) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-5) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * Math.cos(k)) / (k * k));
	} else {
		tmp = 2.0 * (((Math.cos(k) / t) * (l * ((l / k) / k))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.85e-5:
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * math.cos(k)) / (k * k))
	else:
		tmp = 2.0 * (((math.cos(k) / t) * (l * ((l / k) / k))) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.85e-5)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * cos(k)) / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / t) * Float64(l * Float64(Float64(l / k) / k))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.85e-5)
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	else
		tmp = 2.0 * (((cos(k) / t) * (l * ((l / k) / k))) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.85e-5], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(l * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.84999999999999991e-5

   1. Initial program 30.5%

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

      1. associate-/r* 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-/r* 35.5%

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

      4. associate-*r/ 36.7%

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

      5. associate-/l* 35.5%

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

      6. +-commutative 35.5%

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

      7. unpow2 35.5%

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

      8. sqr-neg 35.5%

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

      9. distribute-frac-neg 35.5%

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

      10. distribute-frac-neg 35.5%

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

      11. unpow2 35.5%

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

      12. associate--l+ 41.4%

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

      13. metadata-eval 41.4%

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

      14. +-rgt-identity 41.4%

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

      15. unpow2 41.4%

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

      16. distribute-frac-neg 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in k around inf 63.1%

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

      1. associate-*r* 63.1%

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

      2. associate-/r* 65.5%

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

      3. unpow2 65.5%

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

      4. associate-*l* 65.5%

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

      5. unpow2 65.5%

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

      6. associate-*l* 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in l around 0 65.5%

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

      1. associate-/l* 65.5%

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

      2. associate-/r/ 65.5%

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

      3. associate-/r* 65.1%

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

      4. unpow2 65.1%

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

      5. unpow2 65.1%

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

      6. times-frac 86.9%

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

      7. *-lft-identity 86.9%

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

      8. associate-*l/ 87.0%

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

      9. *-commutative 87.0%

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

      10. associate-*l* 93.6%

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

      11. associate-*r/ 93.7%

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

      12. *-commutative 93.7%

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

      13. *-lft-identity 93.7%

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

   9. Simplified 93.7%

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

   10. Taylor expanded in k around 0 73.4%

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

       1. unpow2 60.5%

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

   12. Simplified 73.4%

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

   if 1.84999999999999991e-5 < k

   1. Initial program 31.3%

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

      1. associate-/r* 31.3%

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

      2. *-commutative 31.3%

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

      3. associate-/r* 33.8%

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

      4. associate-*r/ 33.8%

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

      5. associate-/l* 33.8%

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

      6. +-commutative 33.8%

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

      7. unpow2 33.8%

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

      8. sqr-neg 33.8%

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

      9. distribute-frac-neg 33.8%

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

      10. distribute-frac-neg 33.8%

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

      11. unpow2 33.8%

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

      12. associate--l+ 53.4%

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

      13. metadata-eval 53.4%

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

      14. +-rgt-identity 53.4%

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

      15. unpow2 53.4%

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

      16. distribute-frac-neg 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in k around inf 71.0%

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

      1. associate-*r* 71.0%

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

      2. associate-/r* 71.0%

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

      3. unpow2 71.0%

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

      4. associate-*l* 71.0%

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

      5. unpow2 71.0%

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

      6. associate-*l* 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in l around 0 71.0%

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

      1. times-frac 71.3%

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

      2. unpow2 71.3%

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

      3. unpow2 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in l around 0 71.3%

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

       1. *-rgt-identity 71.3%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot 1}}{{k}^{2}} \cdot \frac{\cos k}{t}}{{\sin k}^{2}} \]

       2. associate-*r/ 71.3%

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

       3. unpow2 71.3%

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

       4. unpow2 71.3%

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

       5. associate-*l* 78.9%

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

       6. unpow2 78.9%

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

       7. associate-*r/ 78.9%

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

       8. *-rgt-identity 78.9%

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

       9. unpow2 78.9%

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

       10. associate-/r* 90.4%

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

   12. Simplified 90.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k}\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 82.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e-5)
   (* 2.0 (/ (* (* (/ l k) (/ (/ l k) t)) (cos k)) (* k k)))
   (* 2.0 (/ (* (* (/ l k) (/ l k)) (/ (cos k) t)) (pow (sin k) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-5) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) * (cos(k) / t)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.95d-5) then
        tmp = 2.0d0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k))
    else
        tmp = 2.0d0 * ((((l / k) * (l / k)) * (cos(k) / t)) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-5) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * Math.cos(k)) / (k * k));
	} else {
		tmp = 2.0 * ((((l / k) * (l / k)) * (Math.cos(k) / t)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.95e-5:
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * math.cos(k)) / (k * k))
	else:
		tmp = 2.0 * ((((l / k) * (l / k)) * (math.cos(k) / t)) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e-5)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * cos(k)) / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / t)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e-5)
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	else
		tmp = 2.0 * ((((l / k) * (l / k)) * (cos(k) / t)) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.95e-5], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.95e-5

   1. Initial program 30.5%

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

      1. associate-/r* 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-/r* 35.5%

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

      4. associate-*r/ 36.7%

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

      5. associate-/l* 35.5%

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

      6. +-commutative 35.5%

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

      7. unpow2 35.5%

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

      8. sqr-neg 35.5%

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

      9. distribute-frac-neg 35.5%

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

      10. distribute-frac-neg 35.5%

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

      11. unpow2 35.5%

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

      12. associate--l+ 41.4%

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

      13. metadata-eval 41.4%

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

      14. +-rgt-identity 41.4%

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

      15. unpow2 41.4%

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

      16. distribute-frac-neg 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in k around inf 63.1%

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

      1. associate-*r* 63.1%

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

      2. associate-/r* 65.5%

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

      3. unpow2 65.5%

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

      4. associate-*l* 65.5%

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

      5. unpow2 65.5%

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

      6. associate-*l* 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in l around 0 65.5%

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

      1. associate-/l* 65.5%

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

      2. associate-/r/ 65.5%

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

      3. associate-/r* 65.1%

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

      4. unpow2 65.1%

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

      5. unpow2 65.1%

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

      6. times-frac 86.9%

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

      7. *-lft-identity 86.9%

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

      8. associate-*l/ 87.0%

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

      9. *-commutative 87.0%

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

      10. associate-*l* 93.6%

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

      11. associate-*r/ 93.7%

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

      12. *-commutative 93.7%

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

      13. *-lft-identity 93.7%

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

   9. Simplified 93.7%

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

   10. Taylor expanded in k around 0 73.4%

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

       1. unpow2 60.5%

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

   12. Simplified 73.4%

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

   if 1.95e-5 < k

   1. Initial program 31.3%

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

      1. associate-/r* 31.3%

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

      2. *-commutative 31.3%

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

      3. associate-/r* 33.8%

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

      4. associate-*r/ 33.8%

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

      5. associate-/l* 33.8%

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

      6. +-commutative 33.8%

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

      7. unpow2 33.8%

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

      8. sqr-neg 33.8%

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

      9. distribute-frac-neg 33.8%

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

      10. distribute-frac-neg 33.8%

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

      11. unpow2 33.8%

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

      12. associate--l+ 53.4%

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

      13. metadata-eval 53.4%

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

      14. +-rgt-identity 53.4%

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

      15. unpow2 53.4%

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

      16. distribute-frac-neg 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in k around inf 71.0%

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

      1. associate-*r* 71.0%

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

      2. associate-/r* 71.0%

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

      3. unpow2 71.0%

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

      4. associate-*l* 71.0%

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

      5. unpow2 71.0%

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

      6. associate-*l* 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in l around 0 71.0%

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

      1. times-frac 71.3%

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

      2. unpow2 71.3%

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

      3. unpow2 71.3%

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

   9. Simplified 71.3%

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

       1. times-frac 92.0%

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

   11. Applied egg-rr 92.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 4: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sin k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.2e-110)
   (* 2.0 (/ (* (* (/ l k) (/ (/ l k) t)) (cos k)) (* k k)))
   (* 2.0 (pow (/ l (* (sin k) (* k (sqrt t)))) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.2e-110) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	} else {
		tmp = 2.0 * pow((l / (sin(k) * (k * sqrt(t)))), 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 (t <= 3.2d-110) then
        tmp = 2.0d0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k))
    else
        tmp = 2.0d0 * ((l / (sin(k) * (k * sqrt(t)))) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.2e-110) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * Math.cos(k)) / (k * k));
	} else {
		tmp = 2.0 * Math.pow((l / (Math.sin(k) * (k * Math.sqrt(t)))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.2e-110:
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * math.cos(k)) / (k * k))
	else:
		tmp = 2.0 * math.pow((l / (math.sin(k) * (k * math.sqrt(t)))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.2e-110)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * cos(k)) / Float64(k * k)));
	else
		tmp = Float64(2.0 * (Float64(l / Float64(sin(k) * Float64(k * sqrt(t)))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.2e-110)
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	else
		tmp = 2.0 * ((l / (sin(k) * (k * sqrt(t)))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.2e-110], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[(N[Sin[k], $MachinePrecision] * N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.20000000000000028e-110

   1. Initial program 31.7%

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

      1. associate-/r* 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-/r* 37.4%

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

      4. associate-*r/ 38.5%

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

      5. associate-/l* 37.4%

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

      6. +-commutative 37.4%

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

      7. unpow2 37.4%

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

      8. sqr-neg 37.4%

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

      9. distribute-frac-neg 37.4%

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

      10. distribute-frac-neg 37.4%

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

      11. unpow2 37.4%

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

      12. associate--l+ 42.7%

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

      13. metadata-eval 42.7%

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

      14. +-rgt-identity 42.7%

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

      15. unpow2 42.7%

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

      16. distribute-frac-neg 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in k around inf 65.5%

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

      1. associate-*r* 65.5%

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

      2. associate-/r* 67.8%

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

      3. unpow2 67.8%

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

      4. associate-*l* 67.8%

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

      5. unpow2 67.8%

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

      6. associate-*l* 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in l around 0 67.8%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 67.8%

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

      3. associate-/r* 67.5%

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

      4. unpow2 67.5%

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

      5. unpow2 67.5%

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

      6. times-frac 87.7%

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

      7. *-lft-identity 87.7%

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

      8. associate-*l/ 87.7%

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

      9. *-commutative 87.7%

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

      10. associate-*l* 96.4%

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

      11. associate-*r/ 96.5%

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

      12. *-commutative 96.5%

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

      13. *-lft-identity 96.5%

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

   9. Simplified 96.5%

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

   10. Taylor expanded in k around 0 73.8%

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

       1. unpow2 61.3%

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

   12. Simplified 73.8%

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

   if 3.20000000000000028e-110 < t

   1. Initial program 28.6%

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

      1. associate-/r* 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-/r* 29.5%

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

      4. associate-*r/ 29.6%

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

      5. associate-/l* 29.5%

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

      6. +-commutative 29.5%

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

      7. unpow2 29.5%

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

      8. sqr-neg 29.5%

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

      9. distribute-frac-neg 29.5%

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

      10. distribute-frac-neg 29.5%

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

      11. unpow2 29.5%

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

      12. associate--l+ 51.0%

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

      13. metadata-eval 51.0%

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

      14. +-rgt-identity 51.0%

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

      15. unpow2 51.0%

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

      16. distribute-frac-neg 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around inf 65.7%

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

      1. associate-*r* 65.8%

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

      2. associate-/r* 65.9%

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

      3. unpow2 65.9%

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

      4. associate-*l* 65.9%

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

      5. unpow2 65.9%

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

      6. associate-*l* 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in k around 0 56.3%

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

      1. expm1-log1p-u 56.0%

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

      2. expm1-udef 54.3%

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

   9. Applied egg-rr 65.8%

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

       1. expm1-def 71.7%

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

       2. expm1-log1p 72.4%

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

       3. associate-/l/ 70.6%

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

   11. Simplified 70.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-110}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{\sin k \cdot \left(k \cdot \sqrt{t}\right)}\right)}^{2}\\ \end{array} \]

Alternative 5: 74.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-112}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k \cdot \sqrt{t}}}{\sin k}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.25e-112)
   (* 2.0 (/ (* (* (/ l k) (/ (/ l k) t)) (cos k)) (* k k)))
   (* 2.0 (pow (/ (/ l (* k (sqrt t))) (sin k)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.25e-112) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	} else {
		tmp = 2.0 * pow(((l / (k * sqrt(t))) / sin(k)), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.25d-112) then
        tmp = 2.0d0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k))
    else
        tmp = 2.0d0 * (((l / (k * sqrt(t))) / sin(k)) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.25e-112) {
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * Math.cos(k)) / (k * k));
	} else {
		tmp = 2.0 * Math.pow(((l / (k * Math.sqrt(t))) / Math.sin(k)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.25e-112:
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * math.cos(k)) / (k * k))
	else:
		tmp = 2.0 * math.pow(((l / (k * math.sqrt(t))) / math.sin(k)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.25e-112)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * cos(k)) / Float64(k * k)));
	else
		tmp = Float64(2.0 * (Float64(Float64(l / Float64(k * sqrt(t))) / sin(k)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.25e-112)
		tmp = 2.0 * ((((l / k) * ((l / k) / t)) * cos(k)) / (k * k));
	else
		tmp = 2.0 * (((l / (k * sqrt(t))) / sin(k)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.25e-112], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(N[(l / N[(k * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.25000000000000011e-112

   1. Initial program 31.7%

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

      1. associate-/r* 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-/r* 37.4%

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

      4. associate-*r/ 38.5%

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

      5. associate-/l* 37.4%

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

      6. +-commutative 37.4%

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

      7. unpow2 37.4%

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

      8. sqr-neg 37.4%

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

      9. distribute-frac-neg 37.4%

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

      10. distribute-frac-neg 37.4%

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

      11. unpow2 37.4%

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

      12. associate--l+ 42.7%

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

      13. metadata-eval 42.7%

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

      14. +-rgt-identity 42.7%

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

      15. unpow2 42.7%

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

      16. distribute-frac-neg 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in k around inf 65.5%

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

      1. associate-*r* 65.5%

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

      2. associate-/r* 67.8%

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

      3. unpow2 67.8%

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

      4. associate-*l* 67.8%

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

      5. unpow2 67.8%

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

      6. associate-*l* 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in l around 0 67.8%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 67.8%

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

      3. associate-/r* 67.5%

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

      4. unpow2 67.5%

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

      5. unpow2 67.5%

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

      6. times-frac 87.7%

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

      7. *-lft-identity 87.7%

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

      8. associate-*l/ 87.7%

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

      9. *-commutative 87.7%

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

      10. associate-*l* 96.4%

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

      11. associate-*r/ 96.5%

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

      12. *-commutative 96.5%

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

      13. *-lft-identity 96.5%

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

   9. Simplified 96.5%

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

   10. Taylor expanded in k around 0 73.8%

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

       1. unpow2 61.3%

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

   12. Simplified 73.8%

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

   if 1.25000000000000011e-112 < t

   1. Initial program 28.6%

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

      1. associate-/r* 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-/r* 29.5%

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

      4. associate-*r/ 29.6%

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

      5. associate-/l* 29.5%

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

      6. +-commutative 29.5%

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

      7. unpow2 29.5%

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

      8. sqr-neg 29.5%

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

      9. distribute-frac-neg 29.5%

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

      10. distribute-frac-neg 29.5%

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

      11. unpow2 29.5%

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

      12. associate--l+ 51.0%

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

      13. metadata-eval 51.0%

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

      14. +-rgt-identity 51.0%

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

      15. unpow2 51.0%

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

      16. distribute-frac-neg 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around inf 65.7%

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

      1. associate-*r* 65.8%

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

      2. associate-/r* 65.9%

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

      3. unpow2 65.9%

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

      4. associate-*l* 65.9%

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

      5. unpow2 65.9%

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

      6. associate-*l* 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in k around 0 56.3%

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

      1. add-sqr-sqrt 56.3%

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

      2. pow2 56.3%

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

      3. sqrt-div 56.3%

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

      4. sqrt-div 56.2%

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

      5. sqrt-prod 39.4%

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

      6. add-sqr-sqrt 67.8%

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

      7. associate-*r* 66.5%

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

      8. sqrt-prod 66.5%

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

      9. sqrt-unprod 47.7%

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

      10. add-sqr-sqrt 68.8%

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

      11. unpow2 68.8%

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

      12. sqrt-prod 42.7%

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

      13. add-sqr-sqrt 72.4%

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

   9. Applied egg-rr 72.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.25 \cdot 10^{-112}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot \cos k}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{k \cdot \sqrt{t}}}{\sin k}\right)}^{2}\\ \end{array} \]

Alternative 6: 73.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. associate-/r* 30.8%

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

   2. *-commutative 30.8%

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

   3. associate-/r* 35.0%

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

   4. associate-*r/ 35.8%

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

   5. associate-/l* 35.0%

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

   6. +-commutative 35.0%

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

   7. unpow2 35.0%

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

   8. sqr-neg 35.0%

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

   9. distribute-frac-neg 35.0%

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

   10. distribute-frac-neg 35.0%

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

   11. unpow2 35.0%

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

   12. associate--l+ 45.2%

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

   13. metadata-eval 45.2%

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

   14. +-rgt-identity 45.2%

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

   15. unpow2 45.2%

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

   16. distribute-frac-neg 45.2%

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

3. Simplified 45.2%

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

4. Taylor expanded in k around inf 65.6%

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

   1. associate-*r* 65.6%

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

   2. associate-/r* 67.2%

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

   3. unpow2 67.2%

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

   4. associate-*l* 67.2%

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

   5. unpow2 67.2%

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

   6. associate-*l* 71.1%

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

6. Simplified 71.1%

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

7. Taylor expanded in k around 0 59.5%

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

   1. associate-/r* 58.9%

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

   2. unpow2 58.9%

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

   3. unpow2 58.9%

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

   4. times-frac 68.2%

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

   5. *-lft-identity 68.2%

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

   6. associate-*l/ 68.3%

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

   7. *-commutative 68.3%

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

   8. associate-*l* 71.2%

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

   9. associate-*r/ 71.2%

      \[\leadsto 2 \cdot \frac{\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 1}{t}}}{{\sin k}^{2}} \]

   10. *-commutative 71.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{t}}{{\sin k}^{2}} \]

   11. *-lft-identity 71.2%

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

9. Simplified 71.2%

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

10. Final simplification 71.2%

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

Alternative 7: 73.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. associate-/r* 30.8%

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

   2. *-commutative 30.8%

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

   3. associate-/r* 35.0%

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

   4. associate-*r/ 35.8%

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

   5. associate-/l* 35.0%

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

   6. +-commutative 35.0%

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

   7. unpow2 35.0%

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

   8. sqr-neg 35.0%

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

   9. distribute-frac-neg 35.0%

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

   10. distribute-frac-neg 35.0%

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

   11. unpow2 35.0%

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

   12. associate--l+ 45.2%

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

   13. metadata-eval 45.2%

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

   14. +-rgt-identity 45.2%

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

   15. unpow2 45.2%

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

   16. distribute-frac-neg 45.2%

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

3. Simplified 45.2%

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

4. Taylor expanded in k around inf 65.6%

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

   1. associate-*r* 65.6%

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

   2. associate-/r* 67.2%

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

   3. unpow2 67.2%

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

   4. associate-*l* 67.2%

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

   5. unpow2 67.2%

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

   6. associate-*l* 71.1%

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

6. Simplified 71.1%

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

7. Taylor expanded in l around 0 67.2%

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

   1. associate-/l* 67.2%

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

   2. associate-/r/ 67.2%

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

   3. associate-/r* 67.0%

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

   4. unpow2 67.0%

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

   5. unpow2 67.0%

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

   6. times-frac 88.5%

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

   7. *-lft-identity 88.5%

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

   8. associate-*l/ 88.5%

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

   9. *-commutative 88.5%

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

   10. associate-*l* 95.5%

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

   11. associate-*r/ 95.6%

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

   12. *-commutative 95.6%

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

   13. *-lft-identity 95.6%

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

9. Simplified 95.6%

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

10. Taylor expanded in k around 0 70.8%

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

    1. unpow2 59.0%

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

12. Simplified 70.8%

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

13. Final simplification 70.8%

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

Alternative 8: 67.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-115}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right)\\ \mathbf{elif}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-115)
   (* l (* l (/ 2.0 (* t (pow k 4.0)))))
   (if (<= k 3.55e+44)
     (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
     (* (* (/ l k) (/ (/ l k) t)) -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-115) {
		tmp = l * (l * (2.0 / (t * pow(k, 4.0))));
	} else if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	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 <= 5d-115) then
        tmp = l * (l * (2.0d0 / (t * (k ** 4.0d0))))
    else if (k <= 3.55d+44) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = ((l / k) * ((l / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-115) {
		tmp = l * (l * (2.0 / (t * Math.pow(k, 4.0))));
	} else if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5e-115:
		tmp = l * (l * (2.0 / (t * math.pow(k, 4.0))))
	elif k <= 3.55e+44:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-115)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(t * (k ^ 4.0)))));
	elseif (k <= 3.55e+44)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-115)
		tmp = l * (l * (2.0 / (t * (k ^ 4.0))));
	elseif (k <= 3.55e+44)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5e-115], N[(l * N[(l * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.55e+44], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.0000000000000003e-115

   1. Initial program 31.7%

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

      1. associate-/r* 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-/r* 36.9%

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

      4. associate-*r/ 38.3%

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

      5. associate-/l* 36.9%

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

      6. +-commutative 36.9%

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

      7. unpow2 36.9%

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

      8. sqr-neg 36.9%

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

      9. distribute-frac-neg 36.9%

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

      10. distribute-frac-neg 36.9%

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

      11. unpow2 36.9%

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

      12. associate--l+ 42.7%

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

      13. metadata-eval 42.7%

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

      14. +-rgt-identity 42.7%

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

      15. unpow2 42.7%

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

      16. distribute-frac-neg 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in k around 0 21.7%

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

      1. fma-def 21.7%

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

      2. associate-/r* 21.6%

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

      3. unpow2 21.6%

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

      4. unpow2 21.6%

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

      5. associate-*r/ 21.6%

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

      6. unpow2 21.6%

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

      7. *-commutative 21.6%

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

   6. Simplified 21.6%

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

   7. Taylor expanded in l around 0 24.5%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 24.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. associate-*l* 27.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. sub-neg 27.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right) \]

      4. associate-*r/ 27.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      5. metadata-eval 27.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      6. *-commutative 27.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      7. associate-*r/ 27.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right)\right) \]

      8. metadata-eval 27.6%

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

      9. distribute-neg-frac 27.6%

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

      10. metadata-eval 27.6%

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

      11. unpow2 27.6%

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

      12. associate-*r* 34.2%

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

   9. Simplified 34.2%

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

   10. Taylor expanded in k around 0 60.4%

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

   if 5.0000000000000003e-115 < k < 3.55e44

   1. Initial program 27.8%

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

      1. associate-/r* 27.8%

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

      2. *-commutative 27.8%

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

      3. associate-/r* 30.0%

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

      4. associate-*r/ 30.0%

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

      5. associate-/l* 29.9%

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

      6. +-commutative 29.9%

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

      7. unpow2 29.9%

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

      8. sqr-neg 29.9%

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

      9. distribute-frac-neg 29.9%

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

      10. distribute-frac-neg 29.9%

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

      11. unpow2 29.9%

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

      12. associate--l+ 43.8%

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

      13. metadata-eval 43.8%

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

      14. +-rgt-identity 43.8%

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

      15. unpow2 43.8%

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

      16. distribute-frac-neg 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in k around inf 83.5%

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

      1. associate-*r* 83.5%

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

      2. associate-/r* 83.8%

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

      3. unpow2 83.8%

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

      4. associate-*l* 83.8%

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

      5. unpow2 83.8%

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

      6. associate-*l* 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in k around 0 70.2%

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

      1. unpow2 70.2%

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

      2. *-commutative 70.2%

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

      3. times-frac 80.6%

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

   9. Simplified 80.6%

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

   if 3.55e44 < k

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-/r* 34.2%

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

      4. associate-*r/ 34.2%

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

      5. associate-/l* 34.2%

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

      6. +-commutative 34.2%

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

      7. unpow2 34.2%

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

      8. sqr-neg 34.2%

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

      9. distribute-frac-neg 34.2%

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

      10. distribute-frac-neg 34.2%

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

      11. unpow2 34.2%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around 0 49.9%

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

      1. fma-def 49.9%

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

      2. associate-/r* 49.9%

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

      3. unpow2 49.9%

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

      4. unpow2 49.9%

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

      5. associate-*r/ 49.9%

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

      6. unpow2 49.9%

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

      7. *-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in k around inf 53.1%

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

      1. associate-*r/ 53.1%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

      2. *-commutative 53.1%

         \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      3. associate-/r* 53.1%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t}}{{k}^{2}}} \]

      4. associate-*r/ 53.1%

         \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]

      5. *-commutative 53.1%

         \[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.3333333333333333}}{{k}^{2}} \]

      6. associate-*l/ 53.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot -0.3333333333333333} \]

      7. associate-/r* 53.1%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}} \cdot -0.3333333333333333 \]

      8. *-lft-identity 53.1%

         \[\leadsto \frac{\color{blue}{1 \cdot {\ell}^{2}}}{t \cdot {k}^{2}} \cdot -0.3333333333333333 \]

      9. times-frac 53.2%

         \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \cdot -0.3333333333333333 \]

      10. unpow2 53.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot -0.3333333333333333 \]

      11. unpow2 53.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot -0.3333333333333333 \]

      12. times-frac 63.1%

          \[\leadsto \left(\frac{1}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot -0.3333333333333333 \]

      13. *-commutative 63.1%

          \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t}\right)} \cdot -0.3333333333333333 \]

      14. associate-*l* 63.4%

          \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t}\right)\right)} \cdot -0.3333333333333333 \]

      15. associate-*r/ 63.4%

          \[\leadsto \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 1}{t}}\right) \cdot -0.3333333333333333 \]

      16. *-commutative 63.4%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

      17. *-lft-identity 63.4%

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

   9. Simplified 63.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-115}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right)\\ \mathbf{elif}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 9: 67.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+41}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \left(2 \cdot {k}^{-4}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left|\frac{-0.3333333333333333}{k \cdot \frac{k}{\frac{\ell}{t}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9e+41)
   (* l (/ (* l (* 2.0 (pow k -4.0))) t))
   (* l (fabs (/ -0.3333333333333333 (* k (/ k (/ l t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e+41) {
		tmp = l * ((l * (2.0 * pow(k, -4.0))) / t);
	} else {
		tmp = l * fabs((-0.3333333333333333 / (k * (k / (l / t)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9d+41) then
        tmp = l * ((l * (2.0d0 * (k ** (-4.0d0)))) / t)
    else
        tmp = l * abs(((-0.3333333333333333d0) / (k * (k / (l / t)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9e+41) {
		tmp = l * ((l * (2.0 * Math.pow(k, -4.0))) / t);
	} else {
		tmp = l * Math.abs((-0.3333333333333333 / (k * (k / (l / t)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9e+41:
		tmp = l * ((l * (2.0 * math.pow(k, -4.0))) / t)
	else:
		tmp = l * math.fabs((-0.3333333333333333 / (k * (k / (l / t)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9e+41)
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 * (k ^ -4.0))) / t));
	else
		tmp = Float64(l * abs(Float64(-0.3333333333333333 / Float64(k * Float64(k / Float64(l / t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9e+41)
		tmp = l * ((l * (2.0 * (k ^ -4.0))) / t);
	else
		tmp = l * abs((-0.3333333333333333 / (k * (k / (l / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9e+41], N[(l * N[(N[(l * N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(l * N[Abs[N[(-0.3333333333333333 / N[(k * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.0000000000000002e41

   1. Initial program 30.7%

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

      1. associate-/r* 30.7%

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

      2. *-commutative 30.7%

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

      3. associate-/r* 35.2%

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

      4. associate-*r/ 36.3%

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

      5. associate-/l* 35.2%

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

      6. +-commutative 35.2%

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

      7. unpow2 35.2%

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

      8. sqr-neg 35.2%

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

      9. distribute-frac-neg 35.2%

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

      10. distribute-frac-neg 35.2%

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

      11. unpow2 35.2%

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

      12. associate--l+ 43.0%

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

      13. metadata-eval 43.0%

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

      14. +-rgt-identity 43.0%

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

      15. unpow2 43.0%

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

      16. distribute-frac-neg 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in k around 0 24.8%

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

      1. fma-def 24.8%

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

      2. associate-/r* 24.2%

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

      3. unpow2 24.2%

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

      4. unpow2 24.2%

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

      5. associate-*r/ 24.2%

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

      6. unpow2 24.2%

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

      7. *-commutative 24.2%

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

   6. Simplified 24.2%

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

   7. Taylor expanded in l around 0 31.1%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 31.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. associate-*l* 34.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. sub-neg 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right) \]

      4. associate-*r/ 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      5. metadata-eval 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      6. *-commutative 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      7. associate-*r/ 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right)\right) \]

      8. metadata-eval 34.4%

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

      9. distribute-neg-frac 34.4%

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

      10. metadata-eval 34.4%

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

      11. unpow2 34.4%

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

      12. associate-*r* 39.4%

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

   9. Simplified 39.4%

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

   10. Taylor expanded in k around 0 56.9%

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

       1. *-commutative 56.9%

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

       2. *-commutative 56.9%

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

       3. associate-*l/ 56.9%

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

       4. times-frac 56.0%

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

       5. unpow2 56.0%

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

       6. associate-/l* 61.6%

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

   12. Simplified 61.6%

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

       1. associate-*l/ 64.0%

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

       2. div-inv 64.0%

          \[\leadsto \frac{\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4}}\right)}}{\frac{t}{\ell}} \]

       3. pow-flip 64.1%

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

       4. metadata-eval 64.1%

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

   14. Applied egg-rr 64.1%

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

       1. associate-/r/ 65.9%

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

   16. Simplified 65.9%

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

   if 9.0000000000000002e41 < k

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-/r* 34.2%

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

      4. associate-*r/ 34.2%

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

      5. associate-/l* 34.2%

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

      6. +-commutative 34.2%

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

      7. unpow2 34.2%

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

      8. sqr-neg 34.2%

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

      9. distribute-frac-neg 34.2%

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

      10. distribute-frac-neg 34.2%

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

      11. unpow2 34.2%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around 0 49.9%

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

      1. fma-def 49.9%

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

      2. associate-/r* 49.9%

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

      3. unpow2 49.9%

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

      4. unpow2 49.9%

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

      5. associate-*r/ 49.9%

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

      6. unpow2 49.9%

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

      7. *-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in l around 0 53.1%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 53.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. associate-*l* 60.1%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. sub-neg 60.1%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right) \]

      4. associate-*r/ 60.1%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      5. metadata-eval 60.1%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      6. *-commutative 60.1%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      7. associate-*r/ 60.1%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right)\right) \]

      8. metadata-eval 60.1%

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

      9. distribute-neg-frac 60.1%

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

      10. metadata-eval 60.1%

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

      11. unpow2 60.1%

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

      12. associate-*r* 62.5%

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

   9. Simplified 62.5%

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

   10. Taylor expanded in k around inf 60.1%

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

       1. unpow2 60.1%

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

       2. associate-*l* 62.5%

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

   12. Simplified 62.5%

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

       1. add-sqr-sqrt 56.2%

          \[\leadsto \ell \cdot \color{blue}{\left(\sqrt{-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}} \cdot \sqrt{-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}}\right)} \]

       2. sqrt-unprod 58.5%

          \[\leadsto \ell \cdot \color{blue}{\sqrt{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)}} \]

       3. pow2 58.5%

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

       4. *-commutative 58.5%

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

   14. Applied egg-rr 58.5%

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

       1. unpow2 58.5%

          \[\leadsto \ell \cdot \sqrt{\color{blue}{\left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333\right) \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333\right)}} \]

       2. rem-sqrt-square 58.4%

          \[\leadsto \ell \cdot \color{blue}{\left|\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333\right|} \]

       3. *-commutative 58.4%

          \[\leadsto \ell \cdot \left|\color{blue}{-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}}\right| \]

       4. associate-*r/ 58.4%

          \[\leadsto \ell \cdot \left|\color{blue}{\frac{-0.3333333333333333 \cdot \ell}{k \cdot \left(k \cdot t\right)}}\right| \]

       5. associate-/l* 58.4%

          \[\leadsto \ell \cdot \left|\color{blue}{\frac{-0.3333333333333333}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}}\right| \]

       6. associate-*r/ 58.8%

          \[\leadsto \ell \cdot \left|\frac{-0.3333333333333333}{\color{blue}{k \cdot \frac{k \cdot t}{\ell}}}\right| \]

       7. associate-/l* 58.6%

          \[\leadsto \ell \cdot \left|\frac{-0.3333333333333333}{k \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}}\right| \]

   16. Simplified 58.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+41}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \left(2 \cdot {k}^{-4}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left|\frac{-0.3333333333333333}{k \cdot \frac{k}{\frac{\ell}{t}}}\right|\\ \end{array} \]

Alternative 10: 66.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.55e+44)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (* (* (/ l k) (/ (/ l k) t)) -0.3333333333333333)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	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 <= 3.55d+44) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = ((l / k) * ((l / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.55e+44:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.55e+44)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.55e+44)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.55e+44], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.55e44

   1. Initial program 30.7%

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

      1. associate-/r* 30.7%

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

      2. *-commutative 30.7%

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

      3. associate-/r* 35.2%

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

      4. associate-*r/ 36.3%

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

      5. associate-/l* 35.2%

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

      6. +-commutative 35.2%

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

      7. unpow2 35.2%

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

      8. sqr-neg 35.2%

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

      9. distribute-frac-neg 35.2%

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

      10. distribute-frac-neg 35.2%

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

      11. unpow2 35.2%

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

      12. associate--l+ 43.0%

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

      13. metadata-eval 43.0%

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

      14. +-rgt-identity 43.0%

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

      15. unpow2 43.0%

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

      16. distribute-frac-neg 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in k around inf 65.7%

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

      1. associate-*r* 65.7%

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

      2. associate-/r* 67.8%

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

      3. unpow2 67.8%

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

      4. associate-*l* 67.8%

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

      5. unpow2 67.8%

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

      6. associate-*l* 70.9%

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

   6. Simplified 70.9%

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

   7. Taylor expanded in k around 0 56.9%

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

      1. unpow2 56.9%

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

      2. *-commutative 56.9%

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

      3. times-frac 64.5%

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

   9. Simplified 64.5%

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

   if 3.55e44 < k

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-/r* 34.2%

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

      4. associate-*r/ 34.2%

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

      5. associate-/l* 34.2%

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

      6. +-commutative 34.2%

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

      7. unpow2 34.2%

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

      8. sqr-neg 34.2%

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

      9. distribute-frac-neg 34.2%

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

      10. distribute-frac-neg 34.2%

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

      11. unpow2 34.2%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around 0 49.9%

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

      1. fma-def 49.9%

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

      2. associate-/r* 49.9%

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

      3. unpow2 49.9%

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

      4. unpow2 49.9%

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

      5. associate-*r/ 49.9%

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

      6. unpow2 49.9%

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

      7. *-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in k around inf 53.1%

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

      1. associate-*r/ 53.1%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

      2. *-commutative 53.1%

         \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      3. associate-/r* 53.1%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t}}{{k}^{2}}} \]

      4. associate-*r/ 53.1%

         \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]

      5. *-commutative 53.1%

         \[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.3333333333333333}}{{k}^{2}} \]

      6. associate-*l/ 53.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot -0.3333333333333333} \]

      7. associate-/r* 53.1%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}} \cdot -0.3333333333333333 \]

      8. *-lft-identity 53.1%

         \[\leadsto \frac{\color{blue}{1 \cdot {\ell}^{2}}}{t \cdot {k}^{2}} \cdot -0.3333333333333333 \]

      9. times-frac 53.2%

         \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \cdot -0.3333333333333333 \]

      10. unpow2 53.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot -0.3333333333333333 \]

      11. unpow2 53.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot -0.3333333333333333 \]

      12. times-frac 63.1%

          \[\leadsto \left(\frac{1}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot -0.3333333333333333 \]

      13. *-commutative 63.1%

          \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t}\right)} \cdot -0.3333333333333333 \]

      14. associate-*l* 63.4%

          \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t}\right)\right)} \cdot -0.3333333333333333 \]

      15. associate-*r/ 63.4%

          \[\leadsto \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 1}{t}}\right) \cdot -0.3333333333333333 \]

      16. *-commutative 63.4%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

      17. *-lft-identity 63.4%

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

   9. Simplified 63.4%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 11: 67.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \left(2 \cdot {k}^{-4}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.55e+44)
   (* l (/ (* l (* 2.0 (pow k -4.0))) t))
   (* (* (/ l k) (/ (/ l k) t)) -0.3333333333333333)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.55e+44) {
		tmp = l * ((l * (2.0 * pow(k, -4.0))) / t);
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	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 <= 3.55d+44) then
        tmp = l * ((l * (2.0d0 * (k ** (-4.0d0)))) / t)
    else
        tmp = ((l / k) * ((l / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.55e+44) {
		tmp = l * ((l * (2.0 * Math.pow(k, -4.0))) / t);
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.55e+44:
		tmp = l * ((l * (2.0 * math.pow(k, -4.0))) / t)
	else:
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.55e+44)
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 * (k ^ -4.0))) / t));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.55e+44)
		tmp = l * ((l * (2.0 * (k ^ -4.0))) / t);
	else
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.55e+44], N[(l * N[(N[(l * N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.55e44

   1. Initial program 30.7%

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

      1. associate-/r* 30.7%

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

      2. *-commutative 30.7%

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

      3. associate-/r* 35.2%

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

      4. associate-*r/ 36.3%

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

      5. associate-/l* 35.2%

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

      6. +-commutative 35.2%

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

      7. unpow2 35.2%

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

      8. sqr-neg 35.2%

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

      9. distribute-frac-neg 35.2%

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

      10. distribute-frac-neg 35.2%

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

      11. unpow2 35.2%

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

      12. associate--l+ 43.0%

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

      13. metadata-eval 43.0%

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

      14. +-rgt-identity 43.0%

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

      15. unpow2 43.0%

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

      16. distribute-frac-neg 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in k around 0 24.8%

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

      1. fma-def 24.8%

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

      2. associate-/r* 24.2%

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

      3. unpow2 24.2%

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

      4. unpow2 24.2%

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

      5. associate-*r/ 24.2%

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

      6. unpow2 24.2%

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

      7. *-commutative 24.2%

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

   6. Simplified 24.2%

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

   7. Taylor expanded in l around 0 31.1%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 31.1%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. associate-*l* 34.4%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. sub-neg 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right) \]

      4. associate-*r/ 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      5. metadata-eval 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      6. *-commutative 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

      7. associate-*r/ 34.4%

         \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right)\right) \]

      8. metadata-eval 34.4%

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

      9. distribute-neg-frac 34.4%

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

      10. metadata-eval 34.4%

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

      11. unpow2 34.4%

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

      12. associate-*r* 39.4%

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

   9. Simplified 39.4%

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

   10. Taylor expanded in k around 0 56.9%

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

       1. *-commutative 56.9%

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

       2. *-commutative 56.9%

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

       3. associate-*l/ 56.9%

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

       4. times-frac 56.0%

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

       5. unpow2 56.0%

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

       6. associate-/l* 61.6%

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

   12. Simplified 61.6%

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

       1. associate-*l/ 64.0%

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

       2. div-inv 64.0%

          \[\leadsto \frac{\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4}}\right)}}{\frac{t}{\ell}} \]

       3. pow-flip 64.1%

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

       4. metadata-eval 64.1%

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

   14. Applied egg-rr 64.1%

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

       1. associate-/r/ 65.9%

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

   16. Simplified 65.9%

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

   if 3.55e44 < k

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-/r* 34.2%

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

      4. associate-*r/ 34.2%

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

      5. associate-/l* 34.2%

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

      6. +-commutative 34.2%

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

      7. unpow2 34.2%

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

      8. sqr-neg 34.2%

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

      9. distribute-frac-neg 34.2%

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

      10. distribute-frac-neg 34.2%

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

      11. unpow2 34.2%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around 0 49.9%

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

      1. fma-def 49.9%

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

      2. associate-/r* 49.9%

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

      3. unpow2 49.9%

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

      4. unpow2 49.9%

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

      5. associate-*r/ 49.9%

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

      6. unpow2 49.9%

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

      7. *-commutative 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in k around inf 53.1%

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

      1. associate-*r/ 53.1%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

      2. *-commutative 53.1%

         \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      3. associate-/r* 53.1%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t}}{{k}^{2}}} \]

      4. associate-*r/ 53.1%

         \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]

      5. *-commutative 53.1%

         \[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.3333333333333333}}{{k}^{2}} \]

      6. associate-*l/ 53.1%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot -0.3333333333333333} \]

      7. associate-/r* 53.1%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}} \cdot -0.3333333333333333 \]

      8. *-lft-identity 53.1%

         \[\leadsto \frac{\color{blue}{1 \cdot {\ell}^{2}}}{t \cdot {k}^{2}} \cdot -0.3333333333333333 \]

      9. times-frac 53.2%

         \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \cdot -0.3333333333333333 \]

      10. unpow2 53.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot -0.3333333333333333 \]

      11. unpow2 53.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot -0.3333333333333333 \]

      12. times-frac 63.1%

          \[\leadsto \left(\frac{1}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot -0.3333333333333333 \]

      13. *-commutative 63.1%

          \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t}\right)} \cdot -0.3333333333333333 \]

      14. associate-*l* 63.4%

          \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t}\right)\right)} \cdot -0.3333333333333333 \]

      15. associate-*r/ 63.4%

          \[\leadsto \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 1}{t}}\right) \cdot -0.3333333333333333 \]

      16. *-commutative 63.4%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

      17. *-lft-identity 63.4%

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

   9. Simplified 63.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \left(2 \cdot {k}^{-4}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 12: 63.9% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7.6e+97)
   (* 2.0 (/ (/ (* l l) (* k (* k t))) (* k k)))
   (* (* (/ l k) (/ (/ l k) t)) -0.3333333333333333)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.6e+97) {
		tmp = 2.0 * (((l * l) / (k * (k * t))) / (k * k));
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	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 <= 7.6d+97) then
        tmp = 2.0d0 * (((l * l) / (k * (k * t))) / (k * k))
    else
        tmp = ((l / k) * ((l / k) / t)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.6e+97) {
		tmp = 2.0 * (((l * l) / (k * (k * t))) / (k * k));
	} else {
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7.6e+97:
		tmp = 2.0 * (((l * l) / (k * (k * t))) / (k * k))
	else:
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7.6e+97)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * Float64(k * t))) / Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7.6e+97)
		tmp = 2.0 * (((l * l) / (k * (k * t))) / (k * k));
	else
		tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7.6e+97], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.60000000000000071e97

   1. Initial program 30.0%

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

      1. associate-/r* 30.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. *-commutative 30.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-/r* 34.3%

         \[\leadsto \frac{\frac{2}{\left(\sin k \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-*r/ 35.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. associate-/l* 34.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      6. +-commutative 34.3%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      7. unpow2 34.3%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      8. sqr-neg 34.3%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      9. distribute-frac-neg 34.3%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      10. distribute-frac-neg 34.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      11. unpow2 34.3%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      12. associate--l+ 42.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      13. metadata-eval 42.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 42.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      15. unpow2 42.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      16. distribute-frac-neg 42.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

   3. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 67.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 67.1%

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      2. associate-/r* 69.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot t}}{{\sin k}^{2}}} \]

      3. unpow2 69.1%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot t}}{{\sin k}^{2}} \]

      4. associate-*l* 69.1%

         \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot t}}{{\sin k}^{2}} \]

      5. unpow2 69.1%

         \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t}}{{\sin k}^{2}} \]

      6. associate-*l* 72.0%

         \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}}}{{\sin k}^{2}} \]

   6. Simplified 72.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{k \cdot \left(k \cdot t\right)}}{{\sin k}^{2}}} \]

   7. Taylor expanded in k around 0 62.0%

      \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \color{blue}{\ell}}{k \cdot \left(k \cdot t\right)}}{{\sin k}^{2}} \]

   8. Taylor expanded in k around 0 60.3%

      \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}}{\color{blue}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 60.3%

         \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}}{\color{blue}{k \cdot k}} \]

   10. Simplified 60.3%

       \[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}}{\color{blue}{k \cdot k}} \]

   if 7.60000000000000071e97 < k

   1. Initial program 33.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 33.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. *-commutative 33.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-/r* 37.7%

         \[\leadsto \frac{\frac{2}{\left(\sin k \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-*r/ 37.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. associate-/l* 37.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      6. +-commutative 37.7%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      7. unpow2 37.7%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      8. sqr-neg 37.7%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      9. distribute-frac-neg 37.7%

         \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

      10. distribute-frac-neg 37.7%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

      11. unpow2 37.7%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

      12. associate--l+ 57.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

      13. metadata-eval 57.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 57.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

      15. unpow2 57.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

      16. distribute-frac-neg 57.5%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

   3. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 54.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. fma-def 54.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

      2. associate-/r* 54.2%

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      3. unpow2 54.2%

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      4. unpow2 54.2%

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

      5. associate-*r/ 54.2%

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      6. unpow2 54.2%

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t}\right) \]

      7. *-commutative 54.2%

         \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   6. Simplified 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{4}}\right)} \]

   7. Taylor expanded in k around inf 56.2%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 56.2%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

      2. *-commutative 56.2%

         \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

      3. associate-/r* 56.0%

         \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t}}{{k}^{2}}} \]

      4. associate-*r/ 56.0%

         \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]

      5. *-commutative 56.0%

         \[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.3333333333333333}}{{k}^{2}} \]

      6. associate-*l/ 56.0%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot -0.3333333333333333} \]

      7. associate-/r* 56.2%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}} \cdot -0.3333333333333333 \]

      8. *-lft-identity 56.2%

         \[\leadsto \frac{\color{blue}{1 \cdot {\ell}^{2}}}{t \cdot {k}^{2}} \cdot -0.3333333333333333 \]

      9. times-frac 56.2%

         \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \cdot -0.3333333333333333 \]

      10. unpow2 56.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot -0.3333333333333333 \]

      11. unpow2 56.2%

          \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot -0.3333333333333333 \]

      12. times-frac 67.8%

          \[\leadsto \left(\frac{1}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot -0.3333333333333333 \]

      13. *-commutative 67.8%

          \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t}\right)} \cdot -0.3333333333333333 \]

      14. associate-*l* 68.2%

          \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t}\right)\right)} \cdot -0.3333333333333333 \]

      15. associate-*r/ 68.2%

          \[\leadsto \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 1}{t}}\right) \cdot -0.3333333333333333 \]

      16. *-commutative 68.2%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

      17. *-lft-identity 68.2%

          \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{\frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

   9. Simplified 68.2%

      \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 13: 34.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* l (* -0.3333333333333333 (/ l (* k (* k t))))))

C

double code(double t, double l, double k) {
	return l * (-0.3333333333333333 * (l / (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 * ((-0.3333333333333333d0) * (l / (k * (k * t))))
end function

Java

public static double code(double t, double l, double k) {
	return l * (-0.3333333333333333 * (l / (k * (k * t))));
}

Python

def code(t, l, k):
	return l * (-0.3333333333333333 * (l / (k * (k * t))))

Julia

function code(t, l, k)
	return Float64(l * Float64(-0.3333333333333333 * Float64(l / Float64(k * Float64(k * t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (-0.3333333333333333 * (l / (k * (k * t))));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(-0.3333333333333333 * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 30.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-/r* 35.0%

      \[\leadsto \frac{\frac{2}{\left(\sin k \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-*r/ 35.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. associate-/l* 35.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   6. +-commutative 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   7. unpow2 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   8. sqr-neg 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   9. distribute-frac-neg 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

   10. distribute-frac-neg 35.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

   11. unpow2 35.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

   12. associate--l+ 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

   13. metadata-eval 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

   15. unpow2 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

3. Simplified 45.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around 0 30.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. fma-def 30.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. associate-/r* 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. associate-*r/ 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

   6. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t}\right) \]

   7. *-commutative 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}}\right) \]

6. Simplified 30.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{4}}\right)} \]

7. Taylor expanded in l around 0 36.4%

   \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. unpow2 36.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   2. associate-*l* 40.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

   3. sub-neg 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right) \]

   4. associate-*r/ 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

   5. metadata-eval 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

   6. *-commutative 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

   7. associate-*r/ 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right)\right) \]

   8. metadata-eval 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\frac{\color{blue}{0.3333333333333333}}{{k}^{2} \cdot t}\right)\right)\right) \]

   9. distribute-neg-frac 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}}\right)\right) \]

   10. metadata-eval 40.6%

       \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{\color{blue}{-0.3333333333333333}}{{k}^{2} \cdot t}\right)\right) \]

   11. unpow2 40.6%

       \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right)\right) \]

   12. associate-*r* 45.0%

       \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right)\right) \]

9. Simplified 45.0%

   \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

10. Taylor expanded in k around inf 31.5%

    \[\leadsto \ell \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. unpow2 31.5%

       \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    2. associate-*l* 32.3%

       \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

12. Simplified 32.3%

    \[\leadsto \ell \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)} \]

13. Final simplification 32.3%

    \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \]

Alternative 14: 34.6% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{-0.3333333333333333}{k \cdot \frac{k}{\frac{\ell}{t}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* l (/ -0.3333333333333333 (* k (/ k (/ l t))))))

C

double code(double t, double l, double k) {
	return l * (-0.3333333333333333 / (k * (k / (l / 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 * ((-0.3333333333333333d0) / (k * (k / (l / t))))
end function

Java

public static double code(double t, double l, double k) {
	return l * (-0.3333333333333333 / (k * (k / (l / t))));
}

Python

def code(t, l, k):
	return l * (-0.3333333333333333 / (k * (k / (l / t))))

Julia

function code(t, l, k)
	return Float64(l * Float64(-0.3333333333333333 / Float64(k * Float64(k / Float64(l / t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * (-0.3333333333333333 / (k * (k / (l / t))));
end

Wolfram

code[t_, l_, k_] := N[(l * N[(-0.3333333333333333 / N[(k * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 30.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-/r* 35.0%

      \[\leadsto \frac{\frac{2}{\left(\sin k \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-*r/ 35.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. associate-/l* 35.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   6. +-commutative 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   7. unpow2 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   8. sqr-neg 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   9. distribute-frac-neg 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

   10. distribute-frac-neg 35.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

   11. unpow2 35.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

   12. associate--l+ 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

   13. metadata-eval 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

   15. unpow2 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

3. Simplified 45.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around 0 30.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. fma-def 30.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. associate-/r* 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. associate-*r/ 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

   6. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t}\right) \]

   7. *-commutative 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}}\right) \]

6. Simplified 30.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{4}}\right)} \]

7. Taylor expanded in l around 0 36.4%

   \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
8. Step-by-step derivation

   1. unpow2 36.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

   2. associate-*l* 40.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

   3. sub-neg 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)}\right) \]

   4. associate-*r/ 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

   5. metadata-eval 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

   6. *-commutative 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)\right) \]

   7. associate-*r/ 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right)\right) \]

   8. metadata-eval 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \left(-\frac{\color{blue}{0.3333333333333333}}{{k}^{2} \cdot t}\right)\right)\right) \]

   9. distribute-neg-frac 40.6%

      \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}}\right)\right) \]

   10. metadata-eval 40.6%

       \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{\color{blue}{-0.3333333333333333}}{{k}^{2} \cdot t}\right)\right) \]

   11. unpow2 40.6%

       \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right)\right) \]

   12. associate-*r* 45.0%

       \[\leadsto \ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{-0.3333333333333333}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right)\right) \]

9. Simplified 45.0%

   \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{2}{t \cdot {k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

10. Taylor expanded in k around inf 31.5%

    \[\leadsto \ell \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. unpow2 31.5%

       \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    2. associate-*l* 32.3%

       \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

12. Simplified 32.3%

    \[\leadsto \ell \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)} \]

13. Taylor expanded in l around 0 31.5%

    \[\leadsto \ell \cdot \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{{k}^{2} \cdot t}\right)} \]
14. Step-by-step derivation

    1. unpow2 31.5%

       \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

    2. associate-*r* 32.3%

       \[\leadsto \ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]

    3. associate-*r/ 32.3%

       \[\leadsto \ell \cdot \color{blue}{\frac{-0.3333333333333333 \cdot \ell}{k \cdot \left(k \cdot t\right)}} \]

    4. associate-/l* 32.3%

       \[\leadsto \ell \cdot \color{blue}{\frac{-0.3333333333333333}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}} \]

    5. associate-*r/ 32.5%

       \[\leadsto \ell \cdot \frac{-0.3333333333333333}{\color{blue}{k \cdot \frac{k \cdot t}{\ell}}} \]

    6. associate-/l* 32.6%

       \[\leadsto \ell \cdot \frac{-0.3333333333333333}{k \cdot \color{blue}{\frac{k}{\frac{\ell}{t}}}} \]

15. Simplified 32.6%

    \[\leadsto \ell \cdot \color{blue}{\frac{-0.3333333333333333}{k \cdot \frac{k}{\frac{\ell}{t}}}} \]

16. Final simplification 32.6%

    \[\leadsto \ell \cdot \frac{-0.3333333333333333}{k \cdot \frac{k}{\frac{\ell}{t}}} \]

Alternative 15: 34.9% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* (/ l k) (/ (/ l k) t)) -0.3333333333333333))

C

double code(double t, double l, double k) {
	return ((l / k) * ((l / k) / t)) * -0.3333333333333333;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / k) * ((l / k) / t)) * (-0.3333333333333333d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((l / k) * ((l / k) / t)) * -0.3333333333333333;
}

Python

def code(t, l, k):
	return ((l / k) * ((l / k) / t)) * -0.3333333333333333

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / k) * Float64(Float64(l / k) / t)) * -0.3333333333333333)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / k) * ((l / k) / t)) * -0.3333333333333333;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 30.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. *-commutative 30.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-/r* 35.0%

      \[\leadsto \frac{\frac{2}{\left(\sin k \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-*r/ 35.8%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. associate-/l* 35.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}} \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   6. +-commutative 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   7. unpow2 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   8. sqr-neg 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   9. distribute-frac-neg 35.0%

      \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{\frac{-k}{t}} \cdot \left(-\frac{k}{t}\right) + 1\right) - 1} \]

   10. distribute-frac-neg 35.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\frac{-k}{t} \cdot \color{blue}{\frac{-k}{t}} + 1\right) - 1} \]

   11. unpow2 35.0%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\left(\color{blue}{{\left(\frac{-k}{t}\right)}^{2}} + 1\right) - 1} \]

   12. associate--l+ 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2} + \left(1 - 1\right)}} \]

   13. metadata-eval 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{{\left(\frac{-k}{t}\right)}^{2}}} \]

   15. unpow2 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\frac{-k}{t} \cdot \frac{-k}{t}}} \]

   16. distribute-frac-neg 45.2%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{\color{blue}{\left(-\frac{k}{t}\right)} \cdot \frac{-k}{t}} \]

3. Simplified 45.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around 0 30.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. fma-def 30.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. associate-/r* 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. associate-*r/ 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

   6. unpow2 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{4} \cdot t}\right) \]

   7. *-commutative 30.5%

      \[\leadsto \mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{t \cdot {k}^{4}}}\right) \]

6. Simplified 30.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}, \frac{2 \cdot \left(\ell \cdot \ell\right)}{t \cdot {k}^{4}}\right)} \]

7. Taylor expanded in k around inf 28.9%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
8. Step-by-step derivation

   1. associate-*r/ 28.9%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

   2. *-commutative 28.9%

      \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

   3. associate-/r* 28.9%

      \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t}}{{k}^{2}}} \]

   4. associate-*r/ 28.9%

      \[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]

   5. *-commutative 28.9%

      \[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot -0.3333333333333333}}{{k}^{2}} \]

   6. associate-*l/ 28.9%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{k}^{2}} \cdot -0.3333333333333333} \]

   7. associate-/r* 28.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t \cdot {k}^{2}}} \cdot -0.3333333333333333 \]

   8. *-lft-identity 28.9%

      \[\leadsto \frac{\color{blue}{1 \cdot {\ell}^{2}}}{t \cdot {k}^{2}} \cdot -0.3333333333333333 \]

   9. times-frac 29.0%

      \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \cdot -0.3333333333333333 \]

   10. unpow2 29.0%

       \[\leadsto \left(\frac{1}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot -0.3333333333333333 \]

   11. unpow2 29.0%

       \[\leadsto \left(\frac{1}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot -0.3333333333333333 \]

   12. times-frac 32.8%

       \[\leadsto \left(\frac{1}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot -0.3333333333333333 \]

   13. *-commutative 32.8%

       \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t}\right)} \cdot -0.3333333333333333 \]

   14. associate-*l* 32.8%

       \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t}\right)\right)} \cdot -0.3333333333333333 \]

   15. associate-*r/ 32.8%

       \[\leadsto \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot 1}{t}}\right) \cdot -0.3333333333333333 \]

   16. *-commutative 32.8%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

   17. *-lft-identity 32.8%

       \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\color{blue}{\frac{\ell}{k}}}{t}\right) \cdot -0.3333333333333333 \]

9. Simplified 32.8%

   \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333} \]

10. Final simplification 32.8%

    \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.3333333333333333 \]

Reproduce

herbie shell --seed 2023275 
(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))))