Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 95.9%

Time: 23.8s

Alternatives: 12

Speedup: 38.3×

• 34.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 7.2e+92)
   (/ (* l (/ 2.0 k)) (* (sin k) (* (* k t) (/ (tan k) l))))
   (* 2.0 (* (pow (/ l k) 2.0) (/ (/ (cos k) t) (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.2e+92) {
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * ((cos(k) / t) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.2d+92) then
        tmp = (l * (2.0d0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)))
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * ((cos(k) / t) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.2e+92) {
		tmp = (l * (2.0 / k)) / (Math.sin(k) * ((k * t) * (Math.tan(k) / l)));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 7.2e+92:
		tmp = (l * (2.0 / k)) / (math.sin(k) * ((k * t) * (math.tan(k) / l)))
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * ((math.cos(k) / t) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 7.2e+92)
		tmp = Float64(Float64(l * Float64(2.0 / k)) / Float64(sin(k) * Float64(Float64(k * t) * Float64(tan(k) / l))));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7.2e+92)
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * ((cos(k) / t) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 7.2e+92], N[(N[(l * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.2e92

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.2%

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

      4. associate-/r/ 38.2%

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

      5. *-commutative 38.2%

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

      6. times-frac 39.2%

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

      7. +-commutative 39.2%

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

      8. associate--l+ 46.0%

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

      9. metadata-eval 46.0%

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

      10. +-rgt-identity 46.0%

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

      11. times-frac 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 87.6%

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

      2. frac-times 77.4%

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

   8. Applied egg-rr 77.4%

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

      1. times-frac 78.2%

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

      2. associate-*r/ 78.1%

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

      3. associate-*l/ 77.5%

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

      4. associate-/r* 77.4%

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

      5. unpow2 77.4%

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

      6. associate-*r/ 77.5%

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

      7. associate-/r* 77.5%

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

      8. associate-/r* 77.0%

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

      9. unpow2 77.0%

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

   10. Simplified 77.0%

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

       1. expm1-log1p-u 45.0%

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

       2. expm1-udef 41.7%

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

       3. times-frac 41.7%

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

       4. associate-/l/ 41.7%

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

   12. Applied egg-rr 41.7%

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

       1. expm1-def 45.4%

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

       2. expm1-log1p 77.5%

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

       3. associate-/l/ 77.4%

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

       4. *-commutative 77.4%

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

       5. associate-/l* 82.4%

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

   14. Simplified 82.4%

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

       1. expm1-log1p-u 47.3%

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

       2. expm1-udef 43.2%

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

       3. associate-/r/ 43.2%

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

   16. Applied egg-rr 43.2%

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

       1. expm1-def 47.4%

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

       2. expm1-log1p 82.5%

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

       3. associate-*r* 88.0%

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

       4. associate-*r/ 87.9%

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

       5. associate-/r/ 88.4%

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

       6. associate-*l/ 95.1%

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

       7. associate-/r* 95.1%

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

       8. *-commutative 95.1%

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

       9. associate-*l* 96.1%

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

   18. Simplified 96.1%

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

   if 7.2e92 < k

   1. Initial program 39.5%

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

      1. associate-*l* 39.4%

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

      2. associate-*l* 39.4%

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

      3. associate-/r* 39.4%

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

      4. associate-/r/ 39.4%

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

      5. *-commutative 39.4%

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

      6. times-frac 37.0%

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

      7. +-commutative 37.0%

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

      8. associate--l+ 45.0%

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

      9. metadata-eval 45.0%

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

      10. +-rgt-identity 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in t around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. times-frac 67.3%

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

      3. unpow2 67.3%

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

      4. unpow2 67.3%

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

      5. times-frac 95.9%

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

      6. *-commutative 95.9%

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

   6. Simplified 95.9%

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

      1. div-inv 95.9%

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

   8. Applied egg-rr 95.9%

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

   9. Taylor expanded in l around 0 63.3%

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

       1. *-commutative 63.3%

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

       2. times-frac 67.3%

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

       3. *-commutative 67.3%

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

       4. unpow2 67.3%

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

       5. unpow2 67.3%

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

       6. times-frac 95.9%

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

       7. unpow2 95.9%

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

       8. associate-/r* 96.0%

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

   11. Simplified 96.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.1e+93)
   (/ (* l (/ 2.0 k)) (* (sin k) (* (* k t) (/ (tan k) l))))
   (*
    2.0
    (* (* (/ l k) (/ l k)) (* (cos k) (/ 1.0 (* t (pow (sin k) 2.0))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e+93) {
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * (1.0 / (t * pow(sin(k), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d+93) then
        tmp = (l * (2.0d0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) * (1.0d0 / (t * (sin(k) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e+93) {
		tmp = (l * (2.0 / k)) / (Math.sin(k) * ((k * t) * (Math.tan(k) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) * (1.0 / (t * Math.pow(Math.sin(k), 2.0)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.1e+93:
		tmp = (l * (2.0 / k)) / (math.sin(k) * ((k * t) * (math.tan(k) / l)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) * (1.0 / (t * math.pow(math.sin(k), 2.0)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.1e+93)
		tmp = Float64(Float64(l * Float64(2.0 / k)) / Float64(sin(k) * Float64(Float64(k * t) * Float64(tan(k) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) * Float64(1.0 / Float64(t * (sin(k) ^ 2.0))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.1e+93)
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) * (1.0 / (t * (sin(k) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.1e+93], N[(N[(l * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(1.0 / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000011e93

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.2%

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

      4. associate-/r/ 38.2%

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

      5. *-commutative 38.2%

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

      6. times-frac 39.2%

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

      7. +-commutative 39.2%

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

      8. associate--l+ 46.0%

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

      9. metadata-eval 46.0%

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

      10. +-rgt-identity 46.0%

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

      11. times-frac 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 87.6%

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

      2. frac-times 77.4%

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

   8. Applied egg-rr 77.4%

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

      1. times-frac 78.2%

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

      2. associate-*r/ 78.1%

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

      3. associate-*l/ 77.5%

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

      4. associate-/r* 77.4%

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

      5. unpow2 77.4%

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

      6. associate-*r/ 77.5%

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

      7. associate-/r* 77.5%

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

      8. associate-/r* 77.0%

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

      9. unpow2 77.0%

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

   10. Simplified 77.0%

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

       1. expm1-log1p-u 45.0%

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

       2. expm1-udef 41.7%

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

       3. times-frac 41.7%

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

       4. associate-/l/ 41.7%

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

   12. Applied egg-rr 41.7%

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

       1. expm1-def 45.4%

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

       2. expm1-log1p 77.5%

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

       3. associate-/l/ 77.4%

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

       4. *-commutative 77.4%

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

       5. associate-/l* 82.4%

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

   14. Simplified 82.4%

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

       1. expm1-log1p-u 47.3%

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

       2. expm1-udef 43.2%

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

       3. associate-/r/ 43.2%

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

   16. Applied egg-rr 43.2%

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

       1. expm1-def 47.4%

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

       2. expm1-log1p 82.5%

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

       3. associate-*r* 88.0%

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

       4. associate-*r/ 87.9%

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

       5. associate-/r/ 88.4%

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

       6. associate-*l/ 95.1%

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

       7. associate-/r* 95.1%

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

       8. *-commutative 95.1%

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

       9. associate-*l* 96.1%

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

   18. Simplified 96.1%

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

   if 1.10000000000000011e93 < k

   1. Initial program 39.5%

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

      1. associate-*l* 39.4%

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

      2. associate-*l* 39.4%

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

      3. associate-/r* 39.4%

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

      4. associate-/r/ 39.4%

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

      5. *-commutative 39.4%

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

      6. times-frac 37.0%

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

      7. +-commutative 37.0%

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

      8. associate--l+ 45.0%

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

      9. metadata-eval 45.0%

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

      10. +-rgt-identity 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in t around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. times-frac 67.3%

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

      3. unpow2 67.3%

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

      4. unpow2 67.3%

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

      5. times-frac 95.9%

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

      6. *-commutative 95.9%

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

   6. Simplified 95.9%

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

      1. div-inv 95.9%

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

   8. Applied egg-rr 95.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right)\\ \end{array} \]

Alternative 3: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.04 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.04e+93)
   (/ (* l (/ 2.0 k)) (* (sin k) (* (* k t) (/ (tan k) l))))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.04e+93) {
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.04d+93) then
        tmp = (l * (2.0d0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.04e+93) {
		tmp = (l * (2.0 / k)) / (Math.sin(k) * ((k * t) * (Math.tan(k) / l)));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.04e+93:
		tmp = (l * (2.0 / k)) / (math.sin(k) * ((k * t) * (math.tan(k) / l)))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.04e+93)
		tmp = Float64(Float64(l * Float64(2.0 / k)) / Float64(sin(k) * Float64(Float64(k * t) * Float64(tan(k) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.04e+93)
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.04e+93], N[(N[(l * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.04e93

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.2%

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

      4. associate-/r/ 38.2%

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

      5. *-commutative 38.2%

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

      6. times-frac 39.2%

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

      7. +-commutative 39.2%

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

      8. associate--l+ 46.0%

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

      9. metadata-eval 46.0%

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

      10. +-rgt-identity 46.0%

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

      11. times-frac 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 87.6%

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

      2. frac-times 77.4%

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

   8. Applied egg-rr 77.4%

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

      1. times-frac 78.2%

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

      2. associate-*r/ 78.1%

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

      3. associate-*l/ 77.5%

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

      4. associate-/r* 77.4%

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

      5. unpow2 77.4%

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

      6. associate-*r/ 77.5%

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

      7. associate-/r* 77.5%

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

      8. associate-/r* 77.0%

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

      9. unpow2 77.0%

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

   10. Simplified 77.0%

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

       1. expm1-log1p-u 45.0%

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

       2. expm1-udef 41.7%

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

       3. times-frac 41.7%

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

       4. associate-/l/ 41.7%

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

   12. Applied egg-rr 41.7%

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

       1. expm1-def 45.4%

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

       2. expm1-log1p 77.5%

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

       3. associate-/l/ 77.4%

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

       4. *-commutative 77.4%

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

       5. associate-/l* 82.4%

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

   14. Simplified 82.4%

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

       1. expm1-log1p-u 47.3%

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

       2. expm1-udef 43.2%

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

       3. associate-/r/ 43.2%

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

   16. Applied egg-rr 43.2%

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

       1. expm1-def 47.4%

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

       2. expm1-log1p 82.5%

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

       3. associate-*r* 88.0%

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

       4. associate-*r/ 87.9%

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

       5. associate-/r/ 88.4%

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

       6. associate-*l/ 95.1%

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

       7. associate-/r* 95.1%

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

       8. *-commutative 95.1%

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

       9. associate-*l* 96.1%

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

   18. Simplified 96.1%

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

   if 1.04e93 < k

   1. Initial program 39.5%

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

      1. associate-*l* 39.4%

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

      2. associate-*l* 39.4%

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

      3. associate-/r* 39.4%

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

      4. associate-/r/ 39.4%

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

      5. *-commutative 39.4%

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

      6. times-frac 37.0%

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

      7. +-commutative 37.0%

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

      8. associate--l+ 45.0%

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

      9. metadata-eval 45.0%

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

      10. +-rgt-identity 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in t around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. times-frac 67.3%

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

      3. unpow2 67.3%

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

      4. unpow2 67.3%

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

      5. times-frac 95.9%

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

      6. *-commutative 95.9%

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

   6. Simplified 95.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.04 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.1e+93)
   (/ (* l (/ 2.0 k)) (* (sin k) (* (* k t) (/ (tan k) l))))
   (/ (* 2.0 (/ (pow (/ l k) 2.0) t)) (* (sin k) (tan k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e+93) {
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	} else {
		tmp = (2.0 * (pow((l / k), 2.0) / t)) / (sin(k) * tan(k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d+93) then
        tmp = (l * (2.0d0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)))
    else
        tmp = (2.0d0 * (((l / k) ** 2.0d0) / t)) / (sin(k) * tan(k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.1e+93) {
		tmp = (l * (2.0 / k)) / (Math.sin(k) * ((k * t) * (Math.tan(k) / l)));
	} else {
		tmp = (2.0 * (Math.pow((l / k), 2.0) / t)) / (Math.sin(k) * Math.tan(k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.1e+93:
		tmp = (l * (2.0 / k)) / (math.sin(k) * ((k * t) * (math.tan(k) / l)))
	else:
		tmp = (2.0 * (math.pow((l / k), 2.0) / t)) / (math.sin(k) * math.tan(k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.1e+93)
		tmp = Float64(Float64(l * Float64(2.0 / k)) / Float64(sin(k) * Float64(Float64(k * t) * Float64(tan(k) / l))));
	else
		tmp = Float64(Float64(2.0 * Float64((Float64(l / k) ^ 2.0) / t)) / Float64(sin(k) * tan(k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.1e+93)
		tmp = (l * (2.0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)));
	else
		tmp = (2.0 * (((l / k) ^ 2.0) / t)) / (sin(k) * tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.1e+93], N[(N[(l * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000011e93

   1. Initial program 38.4%

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

      1. associate-*l* 38.4%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.2%

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

      4. associate-/r/ 38.2%

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

      5. *-commutative 38.2%

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

      6. times-frac 39.2%

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

      7. +-commutative 39.2%

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

      8. associate--l+ 46.0%

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

      9. metadata-eval 46.0%

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

      10. +-rgt-identity 46.0%

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

      11. times-frac 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. associate-*l* 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 87.6%

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

      2. frac-times 77.4%

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

   8. Applied egg-rr 77.4%

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

      1. times-frac 78.2%

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

      2. associate-*r/ 78.1%

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

      3. associate-*l/ 77.5%

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

      4. associate-/r* 77.4%

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

      5. unpow2 77.4%

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

      6. associate-*r/ 77.5%

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

      7. associate-/r* 77.5%

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

      8. associate-/r* 77.0%

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

      9. unpow2 77.0%

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

   10. Simplified 77.0%

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

       1. expm1-log1p-u 45.0%

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

       2. expm1-udef 41.7%

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

       3. times-frac 41.7%

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

       4. associate-/l/ 41.7%

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

   12. Applied egg-rr 41.7%

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

       1. expm1-def 45.4%

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

       2. expm1-log1p 77.5%

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

       3. associate-/l/ 77.4%

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

       4. *-commutative 77.4%

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

       5. associate-/l* 82.4%

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

   14. Simplified 82.4%

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

       1. expm1-log1p-u 47.3%

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

       2. expm1-udef 43.2%

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

       3. associate-/r/ 43.2%

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

   16. Applied egg-rr 43.2%

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

       1. expm1-def 47.4%

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

       2. expm1-log1p 82.5%

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

       3. associate-*r* 88.0%

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

       4. associate-*r/ 87.9%

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

       5. associate-/r/ 88.4%

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

       6. associate-*l/ 95.1%

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

       7. associate-/r* 95.1%

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

       8. *-commutative 95.1%

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

       9. associate-*l* 96.1%

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

   18. Simplified 96.1%

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

   if 1.10000000000000011e93 < k

   1. Initial program 39.5%

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

      1. associate-*l* 39.4%

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

      2. associate-*l* 39.4%

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

      3. associate-/r* 39.4%

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

      4. associate-/r/ 39.4%

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

      5. *-commutative 39.4%

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

      6. times-frac 37.0%

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

      7. +-commutative 37.0%

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

      8. associate--l+ 45.0%

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

      9. metadata-eval 45.0%

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

      10. +-rgt-identity 45.0%

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

      11. times-frac 45.0%

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

   3. Simplified 45.0%

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

   4. Taylor expanded in t around 0 63.4%

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

      1. unpow2 63.4%

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

      2. associate-*l* 69.0%

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

   6. Simplified 69.0%

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

      1. associate-*l/ 69.1%

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

      2. frac-times 69.1%

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

   8. Applied egg-rr 69.1%

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

      1. times-frac 73.2%

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

      2. associate-*r/ 74.5%

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

      3. associate-*l/ 70.5%

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

      4. associate-/r* 69.0%

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

      5. unpow2 69.0%

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

      6. associate-*r/ 69.0%

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

      7. associate-/r* 70.5%

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

      8. associate-/r* 65.7%

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

      9. unpow2 65.7%

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

   10. Simplified 65.7%

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

   11. Taylor expanded in k around 0 63.3%

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

       1. associate-/r* 67.3%

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

       2. unpow2 67.3%

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

       3. unpow2 67.3%

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

       4. times-frac 95.9%

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

       5. unpow2 95.9%

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

   13. Simplified 95.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \left(\left(k \cdot t\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k \cdot \tan k}\\ \end{array} \]

Alternative 5: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-290}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \cos k}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 1.2e-290)
   (/ 2.0 (/ (* k k) (/ (* l (cos k)) (* (* k k) (/ t l)))))
   (* (/ 2.0 (* k (* k t))) (* (/ l (sin k)) (/ l (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.2e-290) {
		tmp = 2.0 / ((k * k) / ((l * cos(k)) / ((k * k) * (t / l))));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l / sin(k)) * (l / tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.2d-290) then
        tmp = 2.0d0 / ((k * k) / ((l * cos(k)) / ((k * k) * (t / l))))
    else
        tmp = (2.0d0 / (k * (k * t))) * ((l / sin(k)) * (l / tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.2e-290) {
		tmp = 2.0 / ((k * k) / ((l * Math.cos(k)) / ((k * k) * (t / l))));
	} else {
		tmp = (2.0 / (k * (k * t))) * ((l / Math.sin(k)) * (l / Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if l <= 1.2e-290:
		tmp = 2.0 / ((k * k) / ((l * math.cos(k)) / ((k * k) * (t / l))))
	else:
		tmp = (2.0 / (k * (k * t))) * ((l / math.sin(k)) * (l / math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (l <= 1.2e-290)
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l * cos(k)) / Float64(Float64(k * k) * Float64(t / l)))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.2e-290)
		tmp = 2.0 / ((k * k) / ((l * cos(k)) / ((k * k) * (t / l))));
	else
		tmp = (2.0 / (k * (k * t))) * ((l / sin(k)) * (l / tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 1.2e-290], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e-290

   1. Initial program 41.4%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-/l* 75.4%

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

      2. unpow2 75.4%

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

      3. *-commutative 75.4%

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

      4. *-commutative 75.4%

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

      5. times-frac 73.4%

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

      6. unpow2 73.4%

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

      7. associate-/l* 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in k around 0 71.1%

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

      1. unpow2 71.1%

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

   7. Simplified 71.1%

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

      1. frac-times 78.7%

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

   9. Applied egg-rr 78.7%

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

   if 1.2e-290 < l

   1. Initial program 35.6%

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

      1. associate-*l* 35.5%

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

      2. associate-*l* 35.6%

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

      3. associate-/r* 35.2%

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

      4. associate-/r/ 35.2%

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

      5. *-commutative 35.2%

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

      6. times-frac 36.7%

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

      7. +-commutative 36.7%

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

      8. associate--l+ 43.4%

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

      9. metadata-eval 43.4%

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

      10. +-rgt-identity 43.4%

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

      11. times-frac 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in t around 0 79.1%

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

      1. unpow2 79.1%

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

      2. associate-*l* 82.3%

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

   6. Simplified 82.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-290}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \cos k}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \]

Alternative 6: 95.3% accurate, 2.0× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ (* l (/ 2.0 k)) (* (sin k) (* (* k t) (/ (tan k) l)))))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * (2.0d0 / k)) / (sin(k) * ((k * t) * (tan(k) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(l * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(k * t), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. associate-*l* 38.6%

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

   2. associate-*l* 38.7%

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

   3. associate-/r* 38.4%

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

   4. associate-/r/ 38.4%

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

   5. *-commutative 38.4%

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

   6. times-frac 38.8%

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

   7. +-commutative 38.8%

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

   8. associate--l+ 45.8%

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

   9. metadata-eval 45.8%

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

   10. +-rgt-identity 45.8%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in t around 0 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l* 83.9%

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

6. Simplified 83.9%

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

   1. associate-*l/ 83.9%

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

   2. frac-times 75.8%

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

8. Applied egg-rr 75.8%

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

   1. times-frac 77.2%

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

   2. associate-*r/ 77.4%

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

   3. associate-*l/ 76.1%

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

   4. associate-/r* 75.8%

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

   5. unpow2 75.8%

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

   6. associate-*r/ 75.8%

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

   7. associate-/r* 76.1%

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

   8. associate-/r* 74.8%

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

   9. unpow2 74.8%

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

10. Simplified 74.8%

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

    1. expm1-log1p-u 48.3%

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

    2. expm1-udef 44.5%

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

    3. times-frac 44.4%

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

    4. associate-/l/ 44.5%

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

12. Applied egg-rr 44.5%

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

    1. expm1-def 49.3%

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

    2. expm1-log1p 76.1%

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

    3. associate-/l/ 76.1%

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

    4. *-commutative 76.1%

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

    5. associate-/l* 80.1%

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

14. Simplified 80.1%

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

    1. expm1-log1p-u 50.8%

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

    2. expm1-udef 45.7%

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

    3. associate-/r/ 45.7%

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

16. Applied egg-rr 45.7%

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

    1. expm1-def 50.8%

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

    2. expm1-log1p 80.1%

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

    3. associate-*r* 85.1%

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

    4. associate-*r/ 85.0%

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

    5. associate-/r/ 85.4%

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

    6. associate-*l/ 93.3%

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

    7. associate-/r* 93.9%

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

    8. *-commutative 93.9%

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

    9. associate-*l* 94.6%

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

18. Simplified 94.6%

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

19. Final simplification 94.6%

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

Alternative 7: 73.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{\cos k}{k \cdot k}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.6e-64)
   (* 2.0 (/ (pow (/ l k) 2.0) (* k (* k t))))
   (/ 2.0 (/ (* k k) (* (/ l (/ t l)) (/ (cos k) (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.6e-64) {
		tmp = 2.0 * (pow((l / k), 2.0) / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / ((l / (t / l)) * (cos(k) / (k * k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.6d-64) then
        tmp = 2.0d0 * (((l / k) ** 2.0d0) / (k * (k * t)))
    else
        tmp = 2.0d0 / ((k * k) / ((l / (t / l)) * (cos(k) / (k * k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.6e-64) {
		tmp = 2.0 * (Math.pow((l / k), 2.0) / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / ((l / (t / l)) * (Math.cos(k) / (k * k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.6e-64:
		tmp = 2.0 * (math.pow((l / k), 2.0) / (k * (k * t)))
	else:
		tmp = 2.0 / ((k * k) / ((l / (t / l)) * (math.cos(k) / (k * k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.6e-64)
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) / Float64(k * Float64(k * t))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l / Float64(t / l)) * Float64(cos(k) / Float64(k * k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.6e-64)
		tmp = 2.0 * (((l / k) ^ 2.0) / (k * (k * t)));
	else
		tmp = 2.0 / ((k * k) / ((l / (t / l)) * (cos(k) / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.6e-64], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5999999999999998e-64

   1. Initial program 43.3%

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

      1. associate-*l* 43.3%

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

      2. associate-*l* 43.3%

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

      3. associate-/r* 43.2%

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

      4. associate-/r/ 43.2%

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

      5. *-commutative 43.2%

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

      6. times-frac 43.9%

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

      7. +-commutative 43.9%

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

      8. associate--l+ 49.9%

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

      9. metadata-eval 49.9%

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

      10. +-rgt-identity 49.9%

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

      11. times-frac 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in t around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. times-frac 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. times-frac 93.0%

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

      6. *-commutative 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in l around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 76.9%

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

      4. unpow2 76.9%

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

      5. unpow2 76.9%

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

      6. times-frac 93.0%

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

      7. unpow2 93.0%

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

      8. associate-/r* 93.0%

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

      9. associate-*r/ 92.9%

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

   9. Simplified 92.9%

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

       1. *-un-lft-identity 92.9%

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

       2. pow2 92.9%

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

       3. associate-/l* 93.0%

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

       4. pow2 93.0%

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

   11. Applied egg-rr 93.0%

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

   12. Taylor expanded in k around 0 81.1%

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

       1. unpow2 81.1%

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

       2. associate-*l* 83.8%

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

   14. Simplified 83.8%

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

   if 3.5999999999999998e-64 < k

   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. Taylor expanded in t around 0 70.4%

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

      1. associate-/l* 71.9%

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

      2. unpow2 71.9%

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

      3. *-commutative 71.9%

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

      4. *-commutative 71.9%

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

      5. times-frac 71.9%

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

      6. unpow2 71.9%

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

      7. associate-/l* 78.0%

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

   4. Simplified 78.0%

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

   5. Taylor expanded in k around 0 63.3%

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

      1. unpow2 63.3%

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

   7. Simplified 63.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}} \cdot \frac{\cos k}{k \cdot k}}}\\ \end{array} \]

Alternative 8: 73.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.46 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \cos k}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.46e-126)
   (* (* (/ l k) (/ l k)) (/ 2.0 (* k (* k t))))
   (/ 2.0 (/ (* k k) (/ (* l (cos k)) (* (* k k) (/ t l)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.46e-126) {
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / ((l * cos(k)) / ((k * k) * (t / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.46d-126) then
        tmp = ((l / k) * (l / k)) * (2.0d0 / (k * (k * t)))
    else
        tmp = 2.0d0 / ((k * k) / ((l * cos(k)) / ((k * k) * (t / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.46e-126) {
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / ((l * Math.cos(k)) / ((k * k) * (t / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.46e-126:
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)))
	else:
		tmp = 2.0 / ((k * k) / ((l * math.cos(k)) / ((k * k) * (t / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.46e-126)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 / Float64(k * Float64(k * t))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l * cos(k)) / Float64(Float64(k * k) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.46e-126)
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	else
		tmp = 2.0 / ((k * k) / ((l * cos(k)) / ((k * k) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.46e-126], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.46000000000000007e-126

   1. Initial program 45.8%

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

      1. associate-*l* 45.8%

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

      2. associate-*l* 45.8%

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

      3. associate-/r* 45.8%

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

      4. associate-/r/ 45.8%

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

      5. *-commutative 45.8%

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

      6. times-frac 46.5%

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

      7. +-commutative 46.5%

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

      8. associate--l+ 53.2%

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

      9. metadata-eval 53.2%

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

      10. +-rgt-identity 53.2%

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

      11. times-frac 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in t around 0 86.4%

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

      1. unpow2 86.4%

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

      2. associate-*l* 90.1%

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

   6. Simplified 90.1%

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

   7. Taylor expanded in k around 0 70.9%

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

      1. unpow2 70.9%

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

      2. unpow2 70.9%

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

      3. times-frac 83.9%

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

   9. Simplified 83.9%

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

   if 1.46000000000000007e-126 < k

   1. Initial program 28.7%

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

   2. Taylor expanded in t around 0 69.8%

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

      1. associate-/l* 72.0%

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

      2. unpow2 72.0%

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

      3. *-commutative 72.0%

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

      4. *-commutative 72.0%

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

      5. times-frac 69.4%

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

      6. unpow2 69.4%

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

      7. associate-/l* 74.5%

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

   4. Simplified 74.5%

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

   5. Taylor expanded in k around 0 62.3%

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

      1. unpow2 62.3%

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

   7. Simplified 62.3%

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

      1. frac-times 69.8%

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

   9. Applied egg-rr 69.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.46 \cdot 10^{-126}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \cos k}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}}\\ \end{array} \]

Alternative 9: 72.2% accurate, 20.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{\frac{t}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.6e-150)
   (* (* (/ l k) (/ l k)) (/ 2.0 (* k (* k t))))
   (/
    2.0
    (/ (* k k) (/ (+ (/ l (* k k)) (* l -0.16666666666666666)) (/ t l))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-150) {
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / (((l / (k * k)) + (l * -0.16666666666666666)) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.6d-150) then
        tmp = ((l / k) * (l / k)) * (2.0d0 / (k * (k * t)))
    else
        tmp = 2.0d0 / ((k * k) / (((l / (k * k)) + (l * (-0.16666666666666666d0))) / (t / l)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-150) {
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / (((l / (k * k)) + (l * -0.16666666666666666)) / (t / l)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.6e-150:
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)))
	else:
		tmp = 2.0 / ((k * k) / (((l / (k * k)) + (l * -0.16666666666666666)) / (t / l)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.6e-150)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 / Float64(k * Float64(k * t))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(Float64(l / Float64(k * k)) + Float64(l * -0.16666666666666666)) / Float64(t / l))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.6e-150)
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	else
		tmp = 2.0 / ((k * k) / (((l / (k * k)) + (l * -0.16666666666666666)) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.6e-150], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.6000000000000003e-150

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

      2. associate-*l* 45.0%

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

      3. associate-/r* 45.0%

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

      4. associate-/r/ 45.0%

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

      5. *-commutative 45.0%

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

      6. times-frac 45.7%

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

      7. +-commutative 45.7%

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

      8. associate--l+ 52.6%

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

      9. metadata-eval 52.6%

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

      10. +-rgt-identity 52.6%

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

      11. times-frac 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in t around 0 86.0%

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

      1. unpow2 86.0%

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

      2. associate-*l* 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in k around 0 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 83.4%

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

   9. Simplified 83.4%

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

   if 6.6000000000000003e-150 < k

   1. Initial program 30.4%

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

   2. Taylor expanded in t around 0 70.9%

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

      1. associate-/l* 73.0%

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

      2. unpow2 73.0%

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

      3. *-commutative 73.0%

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

      4. *-commutative 73.0%

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

      5. times-frac 70.5%

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

      6. unpow2 70.5%

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

      7. associate-/l* 75.4%

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

   4. Simplified 75.4%

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

      1. associate-*l/ 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in k around 0 68.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\left(\frac{\ell}{{k}^{2}} + -0.5 \cdot \ell\right) - -0.3333333333333333 \cdot \ell}}{\frac{t}{\ell}}}} \]
   8. Step-by-step derivation

      1. associate--l+ 68.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\frac{\ell}{{k}^{2}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}}{\frac{t}{\ell}}}} \]

      2. unpow2 68.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{\color{blue}{k \cdot k}} + \left(-0.5 \cdot \ell - -0.3333333333333333 \cdot \ell\right)}{\frac{t}{\ell}}}} \]

      3. distribute-rgt-out-- 68.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{k \cdot k} + \color{blue}{\ell \cdot \left(-0.5 - -0.3333333333333333\right)}}{\frac{t}{\ell}}}} \]

      4. metadata-eval 68.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{k \cdot k} + \ell \cdot \color{blue}{-0.16666666666666666}}{\frac{t}{\ell}}}} \]

   9. Simplified 68.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\color{blue}{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}}{\frac{t}{\ell}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{k \cdot k} + \ell \cdot -0.16666666666666666}{\frac{t}{\ell}}}}\\ \end{array} \]

Alternative 10: 71.6% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell}}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-150)
   (* (* (/ l k) (/ l k)) (/ 2.0 (* k (* k t))))
   (/ 2.0 (/ (* k k) (/ (/ l (* k k)) (/ t l))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-150) {
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / ((l / (k * k)) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-150) then
        tmp = ((l / k) * (l / k)) * (2.0d0 / (k * (k * t)))
    else
        tmp = 2.0d0 / ((k * k) / ((l / (k * k)) / (t / l)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-150) {
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	} else {
		tmp = 2.0 / ((k * k) / ((l / (k * k)) / (t / l)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5e-150:
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)))
	else:
		tmp = 2.0 / ((k * k) / ((l / (k * k)) / (t / l)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-150)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 / Float64(k * Float64(k * t))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(l / Float64(k * k)) / Float64(t / l))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-150)
		tmp = ((l / k) * (l / k)) * (2.0 / (k * (k * t)));
	else
		tmp = 2.0 / ((k * k) / ((l / (k * k)) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-150], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999999e-150

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

      2. associate-*l* 45.0%

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

      3. associate-/r* 45.0%

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

      4. associate-/r/ 45.0%

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

      5. *-commutative 45.0%

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

      6. times-frac 45.7%

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

      7. +-commutative 45.7%

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

      8. associate--l+ 52.6%

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

      9. metadata-eval 52.6%

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

      10. +-rgt-identity 52.6%

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

      11. times-frac 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in t around 0 86.0%

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

      1. unpow2 86.0%

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

      2. associate-*l* 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in k around 0 70.1%

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

      1. unpow2 70.1%

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

      2. unpow2 70.1%

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

      3. times-frac 83.4%

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

   9. Simplified 83.4%

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

   if 4.9999999999999999e-150 < k

   1. Initial program 30.4%

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

   2. Taylor expanded in t around 0 70.9%

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

      1. associate-/l* 73.0%

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

      2. unpow2 73.0%

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

      3. *-commutative 73.0%

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

      4. *-commutative 73.0%

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

      5. times-frac 70.5%

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

      6. unpow2 70.5%

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

      7. associate-/l* 75.4%

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

   4. Simplified 75.4%

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

      1. associate-*l/ 82.5%

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

   6. Applied egg-rr 82.5%

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

   7. Taylor expanded in k around 0 68.0%

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

      1. unpow2 68.0%

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

   9. Simplified 68.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\frac{\ell}{k \cdot k}}{\frac{t}{\ell}}}}\\ \end{array} \]

Alternative 11: 69.9% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (* (/ l k) (/ l k)) (/ 2.0 (* k (* k t)))))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / k) * (l / k)) * (2.0d0 / (k * (k * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. associate-*l* 38.6%

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

   2. associate-*l* 38.7%

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

   3. associate-/r* 38.4%

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

   4. associate-/r/ 38.4%

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

   5. *-commutative 38.4%

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

   6. times-frac 38.8%

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

   7. +-commutative 38.8%

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

   8. associate--l+ 45.8%

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

   9. metadata-eval 45.8%

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

   10. +-rgt-identity 45.8%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in t around 0 80.6%

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

   1. unpow2 80.6%

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

   2. associate-*l* 83.9%

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

6. Simplified 83.9%

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

7. Taylor expanded in k around 0 66.0%

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

   1. unpow2 66.0%

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

   2. unpow2 66.0%

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

   3. times-frac 74.5%

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

9. Simplified 74.5%

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

10. Final simplification 74.5%

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

Alternative 12: 34.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ (* (/ l k) (/ l k)) t) -0.3333333333333333))

C

k = abs(k);
double code(double t, double l, double k) {
	return (((l / k) * (l / k)) / t) * -0.3333333333333333;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (((l / k) * (l / k)) / t) * (-0.3333333333333333d0)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (((l / k) * (l / k)) / t) * -0.3333333333333333;
}

Python

k = abs(k)
def code(t, l, k):
	return (((l / k) * (l / k)) / t) * -0.3333333333333333

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / t) * -0.3333333333333333)
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (((l / k) * (l / k)) / t) * -0.3333333333333333;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 38.6%

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

   1. associate-*l* 38.6%

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

   2. associate-*l* 38.7%

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

   3. associate-/r* 38.4%

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

   4. associate-/r/ 38.4%

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

   5. *-commutative 38.4%

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

   6. times-frac 38.8%

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

   7. +-commutative 38.8%

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

   8. associate--l+ 45.8%

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

   9. metadata-eval 45.8%

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

   10. +-rgt-identity 45.8%

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

   11. times-frac 50.4%

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

3. Simplified 50.4%

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

4. Taylor expanded in k around 0 29.2%

   \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. +-commutative 29.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]

   2. fma-def 29.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]

   3. unpow2 29.2%

      \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   4. *-commutative 29.2%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   5. times-frac 29.4%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   6. times-frac 33.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right)}\right) \]

   7. unpow2 33.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right)\right) \]

   8. unpow2 33.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right)\right) \]

   9. times-frac 36.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right)\right) \]

   10. distribute-rgt-out 36.9%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}{{t}^{2}}\right)\right) \]

   11. metadata-eval 36.9%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{t \cdot \color{blue}{0.16666666666666666}}{{t}^{2}}\right)\right) \]

   12. metadata-eval 36.9%

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

   13. unpow2 36.9%

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

   14. times-frac 44.6%

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

   15. metadata-eval 44.6%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{t}{t} \cdot \frac{\color{blue}{0.16666666666666666}}{t}\right)\right)\right) \]

6. Simplified 44.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\right)} \]

7. Taylor expanded in k around inf 33.9%

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

   1. *-commutative 33.9%

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

   2. associate-/r* 34.0%

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

   3. unpow2 34.0%

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

   4. unpow2 34.0%

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

9. Simplified 34.0%

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

    1. frac-times 34.7%

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

11. Applied egg-rr 34.7%

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

12. Final simplification 34.7%

    \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot -0.3333333333333333 \]

Reproduce

herbie shell --seed 2023217 
(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))))