Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.3% → 81.4%

Time: 14.2s

Alternatives: 6

Speedup: 3.8×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% 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: 81.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 10000000:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+126}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 10000000.0)
   (/ (/ l k) (* (pow t 3.0) (/ k l)))
   (if (<= k 4.1e+126)
     (* l (* l (/ 2.0 (* (tan k) (* k (* k (* t (sin k))))))))
     (/ 2.0 (* (/ t (pow (/ l k) 2.0)) (* (tan k) (sin k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 10000000.0) {
		tmp = (l / k) / (pow(t, 3.0) * (k / l));
	} else if (k <= 4.1e+126) {
		tmp = l * (l * (2.0 / (tan(k) * (k * (k * (t * sin(k)))))));
	} else {
		tmp = 2.0 / ((t / pow((l / k), 2.0)) * (tan(k) * sin(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 <= 10000000.0d0) then
        tmp = (l / k) / ((t ** 3.0d0) * (k / l))
    else if (k <= 4.1d+126) then
        tmp = l * (l * (2.0d0 / (tan(k) * (k * (k * (t * sin(k)))))))
    else
        tmp = 2.0d0 / ((t / ((l / k) ** 2.0d0)) * (tan(k) * sin(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 <= 10000000.0) {
		tmp = (l / k) / (Math.pow(t, 3.0) * (k / l));
	} else if (k <= 4.1e+126) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (k * (k * (t * Math.sin(k)))))));
	} else {
		tmp = 2.0 / ((t / Math.pow((l / k), 2.0)) * (Math.tan(k) * Math.sin(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 10000000.0:
		tmp = (l / k) / (math.pow(t, 3.0) * (k / l))
	elif k <= 4.1e+126:
		tmp = l * (l * (2.0 / (math.tan(k) * (k * (k * (t * math.sin(k)))))))
	else:
		tmp = 2.0 / ((t / math.pow((l / k), 2.0)) * (math.tan(k) * math.sin(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 10000000.0)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) * Float64(k / l)));
	elseif (k <= 4.1e+126)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * Float64(k * Float64(t * sin(k))))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t / (Float64(l / k) ^ 2.0)) * Float64(tan(k) * sin(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 10000000.0)
		tmp = (l / k) / ((t ^ 3.0) * (k / l));
	elseif (k <= 4.1e+126)
		tmp = l * (l * (2.0 / (tan(k) * (k * (k * (t * sin(k)))))));
	else
		tmp = 2.0 / ((t / ((l / k) ^ 2.0)) * (tan(k) * sin(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 10000000.0], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e+126], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1e7

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 48.6%

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

      3. associate-*r* 48.6%

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

      4. +-commutative 48.6%

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

      5. associate-+r+ 48.6%

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

      6. metadata-eval 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in k around 0 49.6%

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

      1. associate-/l* 49.2%

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

      2. unpow2 49.2%

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

      3. unpow2 49.2%

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

      4. associate-*r/ 56.9%

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

      5. times-frac 66.7%

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

   6. Simplified 66.7%

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

      1. expm1-log1p-u 44.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}\right)}\right)\right)} \]

      2. expm1-udef 41.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}\right)}\right)} - 1} \]

      3. associate-/r* 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}}\right)} - 1 \]

      4. metadata-eval 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}\right)} - 1 \]

      5. associate-/r/ 41.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)}}\right)} - 1 \]

   8. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot {t}^{3}\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 46.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot {t}^{3}\right)}\right)\right)} \]

      2. expm1-log1p 69.0%

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

      3. associate-/r* 69.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot {t}^{3}}} \]

      4. *-commutative 69.0%

         \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}} \]

   10. Simplified 69.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{{t}^{3} \cdot \frac{k}{\ell}}} \]

   11. Taylor expanded in k around 0 69.0%

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

   if 1e7 < k < 4.1000000000000001e126

   1. Initial program 35.1%

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

      1. associate-/l/ 35.1%

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

      2. associate-*l/ 35.0%

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

      3. associate-*l/ 35.1%

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

      4. associate-/r/ 35.1%

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

      5. *-commutative 35.1%

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

      6. associate-/l/ 35.1%

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

      7. associate-*r* 35.1%

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

      8. *-commutative 35.1%

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

      9. associate-*r* 35.0%

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

      10. *-commutative 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in k around inf 66.5%

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

      1. unpow2 66.5%

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

      2. *-commutative 66.5%

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

   6. Simplified 66.5%

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

      1. expm1-log1p-u 44.3%

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

      2. expm1-udef 32.1%

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

      3. associate-*l* 40.3%

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

   8. Applied egg-rr 40.3%

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

      1. expm1-def 60.4%

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

      2. expm1-log1p 87.1%

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

      3. associate-*l* 87.1%

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

   10. Simplified 87.1%

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

   if 4.1000000000000001e126 < k

   1. Initial program 53.9%

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

      1. *-commutative 53.9%

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

      2. associate-*l* 53.9%

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

      3. associate-*r* 53.9%

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

      4. +-commutative 53.9%

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

      5. associate-+r+ 53.9%

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

      6. metadata-eval 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around inf 72.1%

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

      1. associate-/l* 72.2%

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

      2. unpow2 72.2%

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

      3. unpow2 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in k around 0 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-/l* 74.6%

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

      3. unpow2 74.6%

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

      4. unpow2 74.6%

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

      5. times-frac 95.0%

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

      6. *-lft-identity 95.0%

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

      7. associate-*l/ 95.1%

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

      8. *-lft-identity 95.1%

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

      9. associate-*l/ 95.0%

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

      10. unpow1 95.0%

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

      11. pow-plus 95.0%

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

      12. associate-*l/ 95.0%

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

      13. *-lft-identity 95.0%

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

      14. metadata-eval 95.0%

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

   9. Simplified 95.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10000000:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+126}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \left(\tan k \cdot \sin k\right)}\\ \end{array} \]

Alternative 2: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 10200000:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 10200000.0)
   (/ (/ l k) (* (pow t 3.0) (/ k l)))
   (* l (* l (/ 2.0 (* (tan k) (* k (* k (* t (sin k))))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 10200000.0) {
		tmp = (l / k) / (pow(t, 3.0) * (k / l));
	} else {
		tmp = l * (l * (2.0 / (tan(k) * (k * (k * (t * sin(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 <= 10200000.0d0) then
        tmp = (l / k) / ((t ** 3.0d0) * (k / l))
    else
        tmp = l * (l * (2.0d0 / (tan(k) * (k * (k * (t * sin(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 <= 10200000.0) {
		tmp = (l / k) / (Math.pow(t, 3.0) * (k / l));
	} else {
		tmp = l * (l * (2.0 / (Math.tan(k) * (k * (k * (t * Math.sin(k)))))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 10200000.0:
		tmp = (l / k) / (math.pow(t, 3.0) * (k / l))
	else:
		tmp = l * (l * (2.0 / (math.tan(k) * (k * (k * (t * math.sin(k)))))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 10200000.0)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) * Float64(k / l)));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(k * Float64(k * Float64(t * sin(k))))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 10200000.0)
		tmp = (l / k) / ((t ^ 3.0) * (k / l));
	else
		tmp = l * (l * (2.0 / (tan(k) * (k * (k * (t * sin(k)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 10200000.0], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.02e7

   1. Initial program 54.7%

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

      1. *-commutative 54.7%

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

      2. associate-*l* 48.6%

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

      3. associate-*r* 48.6%

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

      4. +-commutative 48.6%

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

      5. associate-+r+ 48.6%

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

      6. metadata-eval 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in k around 0 49.6%

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

      1. associate-/l* 49.2%

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

      2. unpow2 49.2%

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

      3. unpow2 49.2%

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

      4. associate-*r/ 56.9%

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

      5. times-frac 66.7%

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

   6. Simplified 66.7%

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

      1. expm1-log1p-u 44.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}\right)}\right)\right)} \]

      2. expm1-udef 41.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}\right)}\right)} - 1} \]

      3. associate-/r* 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}}\right)} - 1 \]

      4. metadata-eval 41.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}\right)} - 1 \]

      5. associate-/r/ 41.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)}}\right)} - 1 \]

   8. Applied egg-rr 41.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot {t}^{3}\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 46.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot {t}^{3}\right)}\right)\right)} \]

      2. expm1-log1p 69.0%

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

      3. associate-/r* 69.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot {t}^{3}}} \]

      4. *-commutative 69.0%

         \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}} \]

   10. Simplified 69.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{{t}^{3} \cdot \frac{k}{\ell}}} \]

   11. Taylor expanded in k around 0 69.0%

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

   if 1.02e7 < k

   1. Initial program 46.9%

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

      1. associate-/l/ 47.0%

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

      2. associate-*l/ 46.9%

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

      3. associate-*l/ 46.9%

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

      4. associate-/r/ 43.9%

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

      5. *-commutative 43.9%

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

      6. associate-/l/ 43.9%

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

      7. associate-*r* 43.9%

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

      8. *-commutative 43.9%

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

      9. associate-*r* 43.9%

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

      10. *-commutative 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in k around inf 70.0%

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

      1. unpow2 70.0%

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

      2. *-commutative 70.0%

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

   6. Simplified 70.0%

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

      1. expm1-log1p-u 60.2%

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

      2. expm1-udef 54.2%

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

      3. associate-*l* 59.2%

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

   8. Applied egg-rr 59.2%

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

      1. expm1-def 68.2%

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

      2. expm1-log1p 79.7%

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

      3. associate-*l* 84.4%

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

   10. Simplified 84.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10200000:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\right)\\ \end{array} \]

Alternative 3: 61.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-55} \lor \neg \left(t \leq 2.15 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.3e-55) (not (<= t 2.15e-72)))
   (* (/ l (pow t 3.0)) (/ l (* k k)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.3e-55) || !(t <= 2.15e-72)) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	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 ((t <= (-1.3d-55)) .or. (.not. (t <= 2.15d-72))) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.3e-55) || !(t <= 2.15e-72)) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.3e-55) or not (t <= 2.15e-72):
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.3e-55) || !(t <= 2.15e-72))
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.3e-55) || ~((t <= 2.15e-72)))
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.3e-55], N[Not[LessEqual[t, 2.15e-72]], $MachinePrecision]], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2999999999999999e-55 or 2.1499999999999999e-72 < t

   1. Initial program 62.3%

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

      1. associate-*l* 62.3%

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

      2. associate-/l/ 62.3%

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

      3. *-commutative 62.3%

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

      4. associate-*r/ 65.5%

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

      5. associate-/l* 62.7%

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

      6. associate-/r/ 54.6%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in k around 0 50.0%

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

      1. unpow2 50.0%

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

      2. *-commutative 50.0%

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

      3. times-frac 57.0%

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

      4. unpow2 57.0%

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

   6. Simplified 57.0%

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

   if -1.2999999999999999e-55 < t < 2.1499999999999999e-72

   1. Initial program 38.5%

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

      1. associate-/l/ 38.5%

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

      2. associate-*l/ 38.5%

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

      3. associate-*l/ 38.5%

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

      4. associate-/r/ 38.4%

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

      5. *-commutative 38.4%

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

      6. associate-/l/ 38.4%

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

      7. associate-*r* 38.4%

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

      8. *-commutative 38.4%

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

      9. associate-*r* 38.4%

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

      10. *-commutative 38.4%

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

   3. Simplified 38.4%

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

   4. Taylor expanded in k around inf 67.4%

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

      1. unpow2 67.4%

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

      2. *-commutative 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in k around 0 56.0%

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

      1. unpow2 56.0%

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

      2. *-commutative 56.0%

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

   9. Simplified 56.0%

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

   10. Taylor expanded in l around 0 56.0%

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

       1. unpow2 56.0%

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

       2. *-commutative 56.0%

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

       3. times-frac 68.7%

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

   12. Simplified 68.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-55} \lor \neg \left(t \leq 2.15 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 4: 61.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5200 \lor \neg \left(t \leq 3.1 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5200.0) (not (<= t 3.1e-47)))
   (/ (* l (/ l (pow t 3.0))) (* k k))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5200.0) || !(t <= 3.1e-47)) {
		tmp = (l * (l / pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	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 ((t <= (-5200.0d0)) .or. (.not. (t <= 3.1d-47))) then
        tmp = (l * (l / (t ** 3.0d0))) / (k * k)
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5200.0) || !(t <= 3.1e-47)) {
		tmp = (l * (l / Math.pow(t, 3.0))) / (k * k);
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -5200.0) or not (t <= 3.1e-47):
		tmp = (l * (l / math.pow(t, 3.0))) / (k * k)
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -5200.0) || !(t <= 3.1e-47))
		tmp = Float64(Float64(l * Float64(l / (t ^ 3.0))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5200.0) || ~((t <= 3.1e-47)))
		tmp = (l * (l / (t ^ 3.0))) / (k * k);
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -5200.0], N[Not[LessEqual[t, 3.1e-47]], $MachinePrecision]], N[(N[(l * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5200 or 3.0999999999999998e-47 < t

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. associate-/l/ 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r/ 64.3%

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

      5. associate-/l* 61.0%

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

      6. associate-/r/ 52.2%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in k around 0 49.8%

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

      1. unpow2 49.8%

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

      2. *-commutative 49.8%

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

      3. times-frac 56.0%

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

      4. unpow2 56.0%

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

   6. Simplified 56.0%

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

      1. associate-*r/ 57.2%

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

   8. Applied egg-rr 57.2%

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

   if -5200 < t < 3.0999999999999998e-47

   1. Initial program 43.4%

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

      1. associate-/l/ 43.4%

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

      2. associate-*l/ 43.3%

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

      3. associate-*l/ 43.3%

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

      4. associate-/r/ 43.3%

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

      5. *-commutative 43.3%

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

      6. associate-/l/ 43.3%

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

      7. associate-*r* 43.3%

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

      8. *-commutative 43.3%

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

      9. associate-*r* 43.3%

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

      10. *-commutative 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in k around inf 68.3%

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

      1. unpow2 68.3%

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

      2. *-commutative 68.3%

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

   6. Simplified 68.3%

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

   7. Taylor expanded in k around 0 56.7%

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

      1. unpow2 56.7%

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

      2. *-commutative 56.7%

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

   9. Simplified 56.7%

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

   10. Taylor expanded in l around 0 56.7%

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

       1. unpow2 56.7%

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

       2. *-commutative 56.7%

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

       3. times-frac 67.9%

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

   12. Simplified 67.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5200 \lor \neg \left(t \leq 3.1 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 5: 67.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-118} \lor \neg \left(t \leq 2.5 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7.2e-118) (not (<= t 2.5e-70)))
   (/ (/ l k) (* (pow t 3.0) (/ k l)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.2e-118) || !(t <= 2.5e-70)) {
		tmp = (l / k) / (pow(t, 3.0) * (k / l));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	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 ((t <= (-7.2d-118)) .or. (.not. (t <= 2.5d-70))) then
        tmp = (l / k) / ((t ** 3.0d0) * (k / l))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7.2e-118) || !(t <= 2.5e-70)) {
		tmp = (l / k) / (Math.pow(t, 3.0) * (k / l));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -7.2e-118) or not (t <= 2.5e-70):
		tmp = (l / k) / (math.pow(t, 3.0) * (k / l))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -7.2e-118) || !(t <= 2.5e-70))
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) * Float64(k / l)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -7.2e-118) || ~((t <= 2.5e-70)))
		tmp = (l / k) / ((t ^ 3.0) * (k / l));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -7.2e-118], N[Not[LessEqual[t, 2.5e-70]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.2000000000000004e-118 or 2.4999999999999999e-70 < t

   1. Initial program 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*l* 54.0%

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

      3. associate-*r* 54.0%

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

      4. +-commutative 54.0%

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

      5. associate-+r+ 54.0%

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

      6. metadata-eval 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in k around 0 49.8%

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

      1. associate-/l* 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. associate-*r/ 56.7%

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

      5. times-frac 67.5%

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

   6. Simplified 67.5%

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

      1. expm1-log1p-u 54.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}\right)}\right)\right)} \]

      2. expm1-udef 50.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}\right)}\right)} - 1} \]

      3. associate-/r* 50.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{2}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}}\right)} - 1 \]

      4. metadata-eval 50.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}\right)} - 1 \]

      5. associate-/r/ 51.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot {t}^{3}\right)}}\right)} - 1 \]

   8. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot {t}^{3}\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot {t}^{3}\right)}\right)\right)} \]

      2. expm1-log1p 70.1%

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

      3. associate-/r* 70.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot {t}^{3}}} \]

      4. *-commutative 70.0%

         \[\leadsto \frac{\frac{1}{\frac{k}{\ell}}}{\color{blue}{{t}^{3} \cdot \frac{k}{\ell}}} \]

   10. Simplified 70.0%

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{k}{\ell}}}{{t}^{3} \cdot \frac{k}{\ell}}} \]

   11. Taylor expanded in k around 0 70.1%

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

   if -7.2000000000000004e-118 < t < 2.4999999999999999e-70

   1. Initial program 37.0%

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

      1. associate-/l/ 37.0%

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

      2. associate-*l/ 37.0%

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

      3. associate-*l/ 37.0%

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

      4. associate-/r/ 37.0%

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

      5. *-commutative 37.0%

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

      6. associate-/l/ 37.0%

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

      7. associate-*r* 37.0%

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

      8. *-commutative 37.0%

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

      9. associate-*r* 37.0%

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

      10. *-commutative 37.0%

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

   3. Simplified 37.0%

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

   4. Taylor expanded in k around inf 70.8%

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

      1. unpow2 70.8%

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

      2. *-commutative 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in k around 0 57.6%

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

      1. unpow2 57.6%

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

      2. *-commutative 57.6%

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

   9. Simplified 57.6%

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

   10. Taylor expanded in l around 0 57.6%

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

       1. unpow2 57.6%

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

       2. *-commutative 57.6%

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

       3. times-frac 70.9%

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

   12. Simplified 70.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-118} \lor \neg \left(t \leq 2.5 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 6: 55.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

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 = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

   1. associate-/l/ 52.8%

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

   2. associate-*l/ 54.8%

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

   3. associate-*l/ 53.2%

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

   4. associate-/r/ 52.4%

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

   5. *-commutative 52.4%

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

   6. associate-/l/ 52.4%

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

   7. associate-*r* 52.4%

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

   8. *-commutative 52.4%

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

   9. associate-*r* 52.4%

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

   10. *-commutative 52.4%

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

3. Simplified 52.4%

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

4. Taylor expanded in k around inf 52.4%

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

   1. unpow2 52.4%

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

   2. *-commutative 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in k around 0 44.4%

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

   1. unpow2 44.4%

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

   2. *-commutative 44.4%

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

9. Simplified 44.4%

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

10. Taylor expanded in l around 0 44.4%

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

    1. unpow2 44.4%

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

    2. *-commutative 44.4%

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

    3. times-frac 51.0%

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

12. Simplified 51.0%

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

13. Final simplification 51.0%

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

Reproduce

herbie shell --seed 2023238 
(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))))