Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 95.0%

Time: 21.3s

Alternatives: 16

Speedup: 28.1×

• 33.6% 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: 34.6% 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: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{+143}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.25e+143)
   (/ (* 2.0 (/ (/ (/ l k) k) (* t (sin k)))) (/ (tan k) l))
   (* 2.0 (/ (* (/ l (/ k l)) (/ (cos k) k)) (* t (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e+143) {
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (tan(k) / l);
	} else {
		tmp = 2.0 * (((l / (k / l)) * (cos(k) / 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 <= 1.25d+143) then
        tmp = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (tan(k) / l)
    else
        tmp = 2.0d0 * (((l / (k / l)) * (cos(k) / 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 <= 1.25e+143) {
		tmp = (2.0 * (((l / k) / k) / (t * Math.sin(k)))) / (Math.tan(k) / l);
	} else {
		tmp = 2.0 * (((l / (k / l)) * (Math.cos(k) / k)) / (t * Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.25e+143:
		tmp = (2.0 * (((l / k) / k) / (t * math.sin(k)))) / (math.tan(k) / l)
	else:
		tmp = 2.0 * (((l / (k / l)) * (math.cos(k) / 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 <= 1.25e+143)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * sin(k)))) / Float64(tan(k) / l));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k / l)) * Float64(cos(k) / 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 <= 1.25e+143)
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (tan(k) / l);
	else
		tmp = 2.0 * (((l / (k / l)) * (cos(k) / 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, 1.25e+143], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000003e143

   1. Initial program 32.8%

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

      1. associate-*l* 32.8%

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

      2. associate-*l* 32.8%

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

      3. associate-/r* 32.8%

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

      4. associate-/r/ 33.3%

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

      5. *-commutative 33.3%

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

      6. times-frac 33.4%

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

      7. +-commutative 33.4%

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

      8. associate--l+ 42.1%

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

      9. metadata-eval 42.1%

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

      10. +-rgt-identity 42.1%

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

      11. times-frac 45.8%

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

   3. Simplified 45.8%

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

      1. associate-*r* 46.8%

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

      2. frac-2neg 46.8%

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

      3. associate-*r/ 46.9%

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

      4. div-inv 46.8%

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

      5. div-inv 46.8%

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

      6. pow-flip 48.2%

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

      7. metadata-eval 48.2%

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

      8. pow-flip 48.2%

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

      9. metadata-eval 48.2%

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

   5. Applied egg-rr 48.2%

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

      1. associate-/l* 48.2%

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

      2. associate-*l* 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in t around 0 86.2%

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

      1. associate-/r* 89.6%

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

      2. unpow2 89.6%

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

      3. *-commutative 89.6%

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

   10. Simplified 89.6%

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

   11. Taylor expanded in l around 0 89.6%

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

       1. unpow2 89.6%

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

       2. associate-/r* 94.9%

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

   13. Simplified 94.9%

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

       1. frac-2neg 94.9%

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

       2. div-inv 94.9%

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

   15. Applied egg-rr 94.9%

       \[\leadsto \frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\color{blue}{\tan k \cdot \frac{1}{\ell}}} \]
   16. Step-by-step derivation

       1. associate-*r/ 94.9%

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

       2. *-rgt-identity 94.9%

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

   17. Simplified 94.9%

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

   if 1.25000000000000003e143 < k

   1. Initial program 47.8%

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

      1. associate-*l* 47.8%

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

      2. associate-*l* 47.8%

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

      3. associate-/r* 47.8%

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

      4. associate-/r/ 45.2%

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

      5. *-commutative 45.2%

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

      6. times-frac 42.9%

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

      7. +-commutative 42.9%

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

      8. associate--l+ 53.4%

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

      9. metadata-eval 53.4%

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

      10. +-rgt-identity 53.4%

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

      11. times-frac 53.4%

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

   3. Simplified 53.4%

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

      1. associate-*r* 54.0%

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

      2. frac-2neg 54.0%

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

      3. associate-*r/ 54.0%

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

      4. div-inv 54.0%

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

      5. div-inv 54.0%

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

      6. pow-flip 54.0%

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

      7. metadata-eval 54.0%

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

      8. pow-flip 54.0%

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

      9. metadata-eval 54.0%

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

   5. Applied egg-rr 54.0%

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

      1. associate-/l* 54.0%

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

      2. associate-*l* 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in t around 0 72.7%

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

      1. associate-/r* 72.7%

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

      2. unpow2 72.7%

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

      3. *-commutative 72.7%

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

   10. Simplified 72.7%

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

   11. Taylor expanded in l around 0 72.0%

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

       1. associate-/r* 74.3%

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

       2. *-commutative 74.3%

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

       3. unpow2 74.3%

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

       4. times-frac 77.8%

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

       5. unpow2 77.8%

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

       6. associate-/l* 95.4%

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

   13. Simplified 95.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{+143}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 2: 89.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-27}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k \cdot \left(k \cdot t\right)}}{\tan k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.85e-27)
   (/ (* 2.0 (/ (/ (/ l k) k) (* t (sin k)))) (/ k l))
   (* l (* (/ l (sin k)) (/ (/ 2.0 (* k (* k t))) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.85e-27) {
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	} else {
		tmp = l * ((l / sin(k)) * ((2.0 / (k * (k * t))) / 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 <= 1.85d-27) then
        tmp = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (k / l)
    else
        tmp = l * ((l / sin(k)) * ((2.0d0 / (k * (k * t))) / 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 <= 1.85e-27) {
		tmp = (2.0 * (((l / k) / k) / (t * Math.sin(k)))) / (k / l);
	} else {
		tmp = l * ((l / Math.sin(k)) * ((2.0 / (k * (k * t))) / Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.85e-27:
		tmp = (2.0 * (((l / k) / k) / (t * math.sin(k)))) / (k / l)
	else:
		tmp = l * ((l / math.sin(k)) * ((2.0 / (k * (k * t))) / math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.85e-27)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * sin(k)))) / Float64(k / l));
	else
		tmp = Float64(l * Float64(Float64(l / sin(k)) * Float64(Float64(2.0 / Float64(k * Float64(k * t))) / tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.85e-27)
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	else
		tmp = l * ((l / sin(k)) * ((2.0 / (k * (k * t))) / tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.85e-27], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85000000000000014e-27

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-/r* 36.1%

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

      4. associate-/r/ 36.6%

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

      5. *-commutative 36.6%

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

      6. times-frac 36.2%

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

      7. +-commutative 36.2%

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

      8. associate--l+ 44.1%

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

      9. metadata-eval 44.1%

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

      10. +-rgt-identity 44.1%

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

      11. times-frac 48.8%

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

   3. Simplified 48.8%

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

      1. associate-*r* 49.5%

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

      2. frac-2neg 49.5%

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

      3. associate-*r/ 49.5%

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

      4. div-inv 49.5%

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

      5. div-inv 49.5%

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

      6. pow-flip 51.2%

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

      7. metadata-eval 51.2%

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

      8. pow-flip 51.2%

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

      9. metadata-eval 51.2%

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

   5. Applied egg-rr 51.2%

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

      1. associate-/l* 51.1%

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

      2. associate-*l* 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in t around 0 83.1%

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

      1. associate-/r* 87.8%

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

      2. unpow2 87.8%

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

      3. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   11. Taylor expanded in l around 0 87.8%

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

       1. unpow2 87.8%

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

       2. associate-/r* 94.3%

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

   13. Simplified 94.3%

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

   14. Taylor expanded in k around 0 81.3%

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

   if 1.85000000000000014e-27 < k

   1. Initial program 32.8%

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

      1. associate-*l* 32.8%

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

      2. associate-*l* 32.8%

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

      3. associate-/r* 32.8%

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

      4. associate-/r/ 31.5%

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

      5. *-commutative 31.5%

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

      6. times-frac 31.7%

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

      7. +-commutative 31.7%

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

      8. associate--l+ 43.0%

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

      9. metadata-eval 43.0%

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

      10. +-rgt-identity 43.0%

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

      11. times-frac 43.0%

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

   3. Simplified 43.0%

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

      1. associate-*r* 44.5%

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

      2. frac-2neg 44.5%

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

      3. associate-*r/ 44.5%

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

      4. div-inv 44.5%

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

      5. div-inv 44.5%

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

      6. pow-flip 44.5%

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

      7. metadata-eval 44.5%

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

      8. pow-flip 44.5%

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

      9. metadata-eval 44.5%

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

   5. Applied egg-rr 44.5%

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

      1. associate-/l* 44.5%

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

      2. associate-*l* 44.4%

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

   7. Simplified 44.4%

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

   8. Taylor expanded in t around 0 86.7%

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

      1. associate-/r* 85.6%

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

      2. unpow2 85.6%

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

      3. *-commutative 85.6%

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

   10. Simplified 85.6%

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

       1. div-inv 85.6%

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

       2. frac-2neg 85.6%

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

   12. Applied egg-rr 85.6%

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

       1. associate-*r/ 85.6%

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

       2. *-rgt-identity 85.6%

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

       3. associate-/r/ 85.6%

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

   14. Simplified 89.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-27}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k \cdot \left(k \cdot t\right)}}{\tan k}\right)\\ \end{array} \]

Alternative 3: 93.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 / k) * (l / (sin(k) * (k * t)))) * (l / tan(k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. *-commutative 35.0%

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

   6. times-frac 34.8%

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

   7. +-commutative 34.8%

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

   8. associate--l+ 43.8%

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

   9. metadata-eval 43.8%

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

   10. +-rgt-identity 43.8%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 82.0%

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

   1. associate-/r* 82.0%

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

   2. unpow2 82.0%

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

   3. associate-/r* 82.4%

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

6. Simplified 82.4%

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

   1. *-commutative 82.4%

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

   2. clear-num 82.4%

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

   3. frac-times 82.5%

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

   4. *-un-lft-identity 82.5%

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

8. Applied egg-rr 82.5%

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

   1. associate-*r/ 85.2%

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

   2. associate-/l/ 88.7%

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

   3. associate-*l/ 87.2%

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

10. Applied egg-rr 87.2%

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

    1. div-inv 87.2%

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

    2. *-commutative 87.2%

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

    3. associate-/l/ 87.2%

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

    4. associate-*r* 84.6%

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

    5. *-commutative 84.6%

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

    6. associate-*l* 87.2%

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

    7. associate-/l* 88.7%

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

12. Applied egg-rr 88.7%

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

    1. associate-*r/ 88.7%

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

    2. *-rgt-identity 88.7%

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

    3. associate-/r/ 88.7%

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

    4. times-frac 86.8%

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

    5. *-commutative 86.8%

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

    6. times-frac 88.7%

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

14. Simplified 92.4%

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

15. Final simplification 92.4%

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

Alternative 4: 92.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 * (((l / k) / k) / (t * sin(k)))) / (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 = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (tan(k) / l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. *-commutative 35.0%

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

   6. times-frac 34.8%

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

   7. +-commutative 34.8%

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

   8. associate--l+ 43.8%

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

   9. metadata-eval 43.8%

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

   10. +-rgt-identity 43.8%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

   1. associate-*r* 47.9%

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

   2. frac-2neg 47.9%

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

   3. associate-*r/ 47.9%

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

   4. div-inv 47.9%

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

   5. div-inv 47.9%

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

   6. pow-flip 49.1%

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

   7. metadata-eval 49.1%

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

   8. pow-flip 49.1%

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

   9. metadata-eval 49.1%

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

5. Applied egg-rr 49.1%

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

   1. associate-/l* 49.1%

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

   2. associate-*l* 48.9%

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

7. Simplified 48.9%

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

8. Taylor expanded in t around 0 84.2%

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

   1. associate-/r* 87.1%

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

   2. unpow2 87.1%

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

   3. *-commutative 87.1%

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

10. Simplified 87.1%

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

11. Taylor expanded in l around 0 87.1%

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

    1. unpow2 87.1%

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

    2. associate-/r* 93.5%

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

13. Simplified 93.5%

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

    1. frac-2neg 93.5%

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

    2. div-inv 93.4%

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

15. Applied egg-rr 93.4%

    \[\leadsto \frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\color{blue}{\tan k \cdot \frac{1}{\ell}}} \]
16. Step-by-step derivation

    1. associate-*r/ 93.5%

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

    2. *-rgt-identity 93.5%

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

17. Simplified 93.5%

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

18. Final simplification 93.5%

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

Alternative 5: 75.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} + 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.55e-13)
   (/ (* 2.0 (/ (/ (/ l k) k) (* t (sin k)))) (/ k l))
   (*
    2.0
    (*
     (/ (cos k) (* k k))
     (+ (/ (/ (* l l) (* k k)) t) (* 0.3333333333333333 (/ l (/ t l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-13) {
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((((l * l) / (k * k)) / t) + (0.3333333333333333 * (l / (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 <= 1.55d-13) then
        tmp = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (k / l)
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((((l * l) / (k * k)) / t) + (0.3333333333333333d0 * (l / (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 <= 1.55e-13) {
		tmp = (2.0 * (((l / k) / k) / (t * Math.sin(k)))) / (k / l);
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((((l * l) / (k * k)) / t) + (0.3333333333333333 * (l / (t / l)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.55e-13:
		tmp = (2.0 * (((l / k) / k) / (t * math.sin(k)))) / (k / l)
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((((l * l) / (k * k)) / t) + (0.3333333333333333 * (l / (t / l)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.55e-13)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * sin(k)))) / Float64(k / l));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(Float64(Float64(l * l) / Float64(k * k)) / t) + Float64(0.3333333333333333 * Float64(l / Float64(t / l))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.55e-13)
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((((l * l) / (k * k)) / t) + (0.3333333333333333 * (l / (t / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.55e-13], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(0.3333333333333333 * N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55e-13

   1. Initial program 35.8%

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

      1. associate-*l* 35.8%

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

      2. associate-*l* 35.8%

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

      3. associate-/r* 35.8%

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

      4. associate-/r/ 36.4%

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

      5. *-commutative 36.4%

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

      6. times-frac 35.9%

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

      7. +-commutative 35.9%

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

      8. associate--l+ 43.7%

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

      9. metadata-eval 43.7%

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

      10. +-rgt-identity 43.7%

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

      11. times-frac 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. Step-by-step derivation

      1. associate-*r* 48.9%

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

      2. frac-2neg 48.9%

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

      3. associate-*r/ 48.9%

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

      4. div-inv 48.9%

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

      5. div-inv 48.9%

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

      6. pow-flip 50.6%

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

      7. metadata-eval 50.6%

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

      8. pow-flip 50.6%

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

      9. metadata-eval 50.6%

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

   5. Applied egg-rr 50.6%

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

      1. associate-/l* 50.6%

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

      2. associate-*l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in t around 0 83.4%

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

      1. associate-/r* 88.1%

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

      2. unpow2 88.1%

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

      3. *-commutative 88.1%

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

   10. Simplified 88.1%

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

   11. Taylor expanded in l around 0 88.1%

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

       1. unpow2 88.1%

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

       2. associate-/r* 94.5%

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

   13. Simplified 94.5%

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

   14. Taylor expanded in k around 0 81.7%

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

   if 1.55e-13 < k

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. associate-*l* 33.2%

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

      3. associate-/r* 33.2%

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

      4. associate-/r/ 31.9%

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

      5. *-commutative 31.9%

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

      6. times-frac 32.0%

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

      7. +-commutative 32.0%

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

      8. associate--l+ 43.9%

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

      9. metadata-eval 43.9%

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

      10. +-rgt-identity 43.9%

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

      11. times-frac 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 82.1%

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

      1. times-frac 82.0%

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

      2. unpow2 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-/r* 82.0%

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

      5. unpow2 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in k around 0 64.7%

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

      1. associate-/r* 64.7%

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

      2. unpow2 64.7%

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

      3. unpow2 64.7%

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

      4. unpow2 64.7%

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

      5. associate-/l* 64.8%

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

   9. Simplified 64.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} + 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)\right)\\ \end{array} \]

Alternative 6: 74.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.8e-5)
   (/ (* 2.0 (/ (/ (/ l k) k) (* t (sin k)))) (/ k l))
   (* 2.0 (* (/ (cos k) (* k k)) (/ (* l l) (* t (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.8e-5) {
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * (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 <= 2.8d-5) then
        tmp = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (k / l)
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l * l) / (t * (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 <= 2.8e-5) {
		tmp = (2.0 * (((l / k) / k) / (t * Math.sin(k)))) / (k / l);
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l * l) / (t * (k * k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.8e-5:
		tmp = (2.0 * (((l / k) / k) / (t * math.sin(k)))) / (k / l)
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l * l) / (t * (k * k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.8e-5)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * sin(k)))) / Float64(k / l));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l * l) / Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.8e-5)
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.8e-5], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.79999999999999996e-5

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-*l* 35.4%

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

      3. associate-/r* 35.4%

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

      4. associate-/r/ 36.0%

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

      5. *-commutative 36.0%

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

      6. times-frac 35.5%

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

      7. +-commutative 35.5%

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

      8. associate--l+ 43.2%

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

      9. metadata-eval 43.2%

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

      10. +-rgt-identity 43.2%

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

      11. times-frac 47.7%

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

   3. Simplified 47.7%

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

      1. associate-*r* 48.4%

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

      2. frac-2neg 48.4%

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

      3. associate-*r/ 48.4%

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

      4. div-inv 48.4%

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

      5. div-inv 48.4%

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

      6. pow-flip 50.0%

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

      7. metadata-eval 50.0%

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

      8. pow-flip 50.0%

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

      9. metadata-eval 50.0%

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

   5. Applied egg-rr 50.0%

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

      1. associate-/l* 50.0%

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

      2. associate-*l* 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in t around 0 83.6%

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

      1. associate-/r* 88.2%

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

      2. unpow2 88.2%

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

      3. *-commutative 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in l around 0 88.2%

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

       1. unpow2 88.2%

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

       2. associate-/r* 94.5%

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

   13. Simplified 94.5%

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

   14. Taylor expanded in k around 0 81.8%

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

   if 2.79999999999999996e-5 < k

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate-*l* 34.1%

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

      3. associate-/r* 34.1%

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

      4. associate-/r/ 32.8%

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

      5. *-commutative 32.8%

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

      6. times-frac 32.9%

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

      7. +-commutative 32.9%

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

      8. associate--l+ 45.1%

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

      9. metadata-eval 45.1%

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

      10. +-rgt-identity 45.1%

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

      11. times-frac 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in t around 0 81.6%

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

      1. times-frac 81.6%

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

      2. unpow2 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-/r* 81.6%

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

      5. unpow2 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in k around 0 61.5%

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

      1. unpow2 61.5%

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

      2. *-commutative 61.5%

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

      3. unpow2 61.5%

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

   9. Simplified 61.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 7: 74.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-5)
   (/ (* 2.0 (/ (/ (/ l k) k) (* t (sin k)))) (/ k l))
   (* 2.0 (* (/ (cos k) (* k k)) (/ (/ (* l l) (* k k)) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-5) {
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (((l * l) / (k * k)) / t));
	}
	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 <= 2.5d-5) then
        tmp = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (k / l)
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (((l * l) / (k * k)) / t))
    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 <= 2.5e-5) {
		tmp = (2.0 * (((l / k) / k) / (t * Math.sin(k)))) / (k / l);
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (((l * l) / (k * k)) / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.5e-5:
		tmp = (2.0 * (((l / k) / k) / (t * math.sin(k)))) / (k / l)
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (((l * l) / (k * k)) / t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-5)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * sin(k)))) / Float64(k / l));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(Float64(l * l) / Float64(k * k)) / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-5)
		tmp = (2.0 * (((l / k) / k) / (t * sin(k)))) / (k / l);
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * (((l * l) / (k * k)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.5e-5], N[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000012e-5

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-*l* 35.4%

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

      3. associate-/r* 35.4%

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

      4. associate-/r/ 36.0%

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

      5. *-commutative 36.0%

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

      6. times-frac 35.5%

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

      7. +-commutative 35.5%

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

      8. associate--l+ 43.2%

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

      9. metadata-eval 43.2%

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

      10. +-rgt-identity 43.2%

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

      11. times-frac 47.7%

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

   3. Simplified 47.7%

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

      1. associate-*r* 48.4%

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

      2. frac-2neg 48.4%

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

      3. associate-*r/ 48.4%

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

      4. div-inv 48.4%

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

      5. div-inv 48.4%

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

      6. pow-flip 50.0%

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

      7. metadata-eval 50.0%

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

      8. pow-flip 50.0%

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

      9. metadata-eval 50.0%

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

   5. Applied egg-rr 50.0%

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

      1. associate-/l* 50.0%

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

      2. associate-*l* 49.8%

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

   7. Simplified 49.8%

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

   8. Taylor expanded in t around 0 83.6%

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

      1. associate-/r* 88.2%

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

      2. unpow2 88.2%

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

      3. *-commutative 88.2%

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

   10. Simplified 88.2%

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

   11. Taylor expanded in l around 0 88.2%

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

       1. unpow2 88.2%

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

       2. associate-/r* 94.5%

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

   13. Simplified 94.5%

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

   14. Taylor expanded in k around 0 81.8%

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

   if 2.50000000000000012e-5 < k

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate-*l* 34.1%

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

      3. associate-/r* 34.1%

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

      4. associate-/r/ 32.8%

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

      5. *-commutative 32.8%

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

      6. times-frac 32.9%

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

      7. +-commutative 32.9%

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

      8. associate--l+ 45.1%

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

      9. metadata-eval 45.1%

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

      10. +-rgt-identity 45.1%

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

      11. times-frac 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in t around 0 81.6%

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

      1. times-frac 81.6%

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

      2. unpow2 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-/r* 81.6%

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

      5. unpow2 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in k around 0 61.5%

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

      1. associate-/r* 61.5%

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

      2. unpow2 61.5%

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

      3. unpow2 61.5%

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

   9. Simplified 61.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \sin k}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 8: 71.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 * ((l / (k * k)) / (t * sin(k)))) / (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 = (2.0d0 * ((l / (k * k)) / (t * sin(k)))) / (k / l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. *-commutative 35.0%

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

   6. times-frac 34.8%

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

   7. +-commutative 34.8%

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

   8. associate--l+ 43.8%

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

   9. metadata-eval 43.8%

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

   10. +-rgt-identity 43.8%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

   1. associate-*r* 47.9%

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

   2. frac-2neg 47.9%

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

   3. associate-*r/ 47.9%

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

   4. div-inv 47.9%

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

   5. div-inv 47.9%

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

   6. pow-flip 49.1%

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

   7. metadata-eval 49.1%

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

   8. pow-flip 49.1%

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

   9. metadata-eval 49.1%

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

5. Applied egg-rr 49.1%

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

   1. associate-/l* 49.1%

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

   2. associate-*l* 48.9%

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

7. Simplified 48.9%

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

8. Taylor expanded in t around 0 84.2%

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

   1. associate-/r* 87.1%

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

   2. unpow2 87.1%

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

   3. *-commutative 87.1%

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

10. Simplified 87.1%

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

11. Taylor expanded in k around 0 73.8%

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

12. Final simplification 73.8%

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

Alternative 9: 72.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 * (((l / k) / k) / (t * sin(k)))) / (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 = (2.0d0 * (((l / k) / k) / (t * sin(k)))) / (k / l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. *-commutative 35.0%

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

   6. times-frac 34.8%

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

   7. +-commutative 34.8%

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

   8. associate--l+ 43.8%

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

   9. metadata-eval 43.8%

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

   10. +-rgt-identity 43.8%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

   1. associate-*r* 47.9%

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

   2. frac-2neg 47.9%

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

   3. associate-*r/ 47.9%

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

   4. div-inv 47.9%

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

   5. div-inv 47.9%

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

   6. pow-flip 49.1%

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

   7. metadata-eval 49.1%

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

   8. pow-flip 49.1%

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

   9. metadata-eval 49.1%

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

5. Applied egg-rr 49.1%

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

   1. associate-/l* 49.1%

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

   2. associate-*l* 48.9%

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

7. Simplified 48.9%

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

8. Taylor expanded in t around 0 84.2%

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

   1. associate-/r* 87.1%

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

   2. unpow2 87.1%

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

   3. *-commutative 87.1%

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

10. Simplified 87.1%

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

11. Taylor expanded in l around 0 87.1%

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

    1. unpow2 87.1%

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

    2. associate-/r* 93.5%

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

13. Simplified 93.5%

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

14. Taylor expanded in k around 0 75.3%

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

15. Final simplification 75.3%

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

Alternative 10: 63.3% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+29}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e+29)
   (* (/ (* l l) (* k k)) (/ 2.0 (* t (* k k))))
   (* (* l l) (/ -0.3333333333333333 (* k (* k t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e+29) {
		tmp = ((l * l) / (k * k)) * (2.0 / (t * (k * k)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	}
	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 <= 1.95d+29) then
        tmp = ((l * l) / (k * k)) * (2.0d0 / (t * (k * k)))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / (k * (k * t)))
    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 <= 1.95e+29) {
		tmp = ((l * l) / (k * k)) * (2.0 / (t * (k * k)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.95e+29:
		tmp = ((l * l) / (k * k)) * (2.0 / (t * (k * k)))
	else:
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e+29)
		tmp = Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(2.0 / Float64(t * Float64(k * k))));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e+29)
		tmp = ((l * l) / (k * k)) * (2.0 / (t * (k * k)));
	else
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.95e+29], N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999984e29

   1. Initial program 34.7%

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

      1. associate-*l* 34.7%

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

      2. associate-*l* 34.7%

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

      3. associate-/r* 34.7%

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

      4. associate-/r/ 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 35.3%

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

      7. +-commutative 35.3%

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

      8. associate--l+ 42.7%

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

      9. metadata-eval 42.7%

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

      10. +-rgt-identity 42.7%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 82.3%

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

      1. associate-/r* 82.4%

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

      2. unpow2 82.4%

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

      3. associate-/r* 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in k around 0 68.8%

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

      1. unpow2 68.8%

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

      2. unpow2 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. *-commutative 68.8%

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

       2. unpow2 68.8%

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

   12. Simplified 68.8%

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

   if 1.94999999999999984e29 < k

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-/r* 36.1%

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

      4. associate-/r/ 34.6%

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

      5. *-commutative 34.6%

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

      6. times-frac 33.3%

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

      7. +-commutative 33.3%

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

      8. associate--l+ 46.9%

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

      9. metadata-eval 46.9%

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

      10. +-rgt-identity 46.9%

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

      11. times-frac 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in t around 0 81.1%

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

      1. associate-/r* 81.1%

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

      2. unpow2 81.1%

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

      3. associate-/r* 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in k around 0 50.2%

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

      1. +-commutative 50.2%

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

      2. fma-def 50.2%

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

      3. unpow2 50.2%

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

      4. *-commutative 50.2%

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

      5. times-frac 50.1%

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

      6. associate-*r/ 50.1%

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

      7. *-commutative 50.1%

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

      8. times-frac 50.3%

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

      9. unpow2 50.3%

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

      10. unpow2 50.3%

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

      11. times-frac 52.3%

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

      12. unpow2 52.3%

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

      13. associate-/l* 52.7%

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

      14. distribute-rgt-out 52.7%

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

      15. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   10. Taylor expanded in l around 0 59.1%

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

       1. unpow2 59.1%

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

       2. associate-*r/ 59.1%

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

       3. metadata-eval 59.1%

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

       4. *-commutative 59.1%

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

       5. associate-*r/ 59.1%

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

       6. metadata-eval 59.1%

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

       7. unpow2 59.1%

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

   12. Simplified 59.1%

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

   13. Taylor expanded in k around inf 59.1%

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

       1. unpow2 59.1%

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

       2. associate-*r* 59.3%

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

   15. Simplified 59.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+29}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 11: 69.2% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+29}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\frac{2}{k}}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e+29)
   (* (* (/ l k) (/ l k)) (/ (/ (/ 2.0 k) k) t))
   (* (* l l) (/ -0.3333333333333333 (* k (* k t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e+29) {
		tmp = ((l / k) * (l / k)) * (((2.0 / k) / k) / t);
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	}
	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 <= 1.95d+29) then
        tmp = ((l / k) * (l / k)) * (((2.0d0 / k) / k) / t)
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / (k * (k * t)))
    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 <= 1.95e+29) {
		tmp = ((l / k) * (l / k)) * (((2.0 / k) / k) / t);
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.95e+29:
		tmp = ((l / k) * (l / k)) * (((2.0 / k) / k) / t)
	else:
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e+29)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(Float64(2.0 / k) / k) / t));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e+29)
		tmp = ((l / k) * (l / k)) * (((2.0 / k) / k) / t);
	else
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.95e+29], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / k), $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999984e29

   1. Initial program 34.7%

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

      1. associate-*l* 34.7%

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

      2. associate-*l* 34.7%

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

      3. associate-/r* 34.7%

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

      4. associate-/r/ 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 35.3%

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

      7. +-commutative 35.3%

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

      8. associate--l+ 42.7%

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

      9. metadata-eval 42.7%

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

      10. +-rgt-identity 42.7%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 82.3%

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

      1. associate-/r* 82.4%

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

      2. unpow2 82.4%

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

      3. associate-/r* 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in k around 0 68.8%

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

      1. unpow2 68.8%

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

      2. unpow2 68.8%

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

   9. Simplified 68.8%

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

       1. times-frac 75.8%

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

   11. Applied egg-rr 75.8%

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

   if 1.94999999999999984e29 < k

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-/r* 36.1%

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

      4. associate-/r/ 34.6%

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

      5. *-commutative 34.6%

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

      6. times-frac 33.3%

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

      7. +-commutative 33.3%

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

      8. associate--l+ 46.9%

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

      9. metadata-eval 46.9%

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

      10. +-rgt-identity 46.9%

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

      11. times-frac 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in t around 0 81.1%

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

      1. associate-/r* 81.1%

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

      2. unpow2 81.1%

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

      3. associate-/r* 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in k around 0 50.2%

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

      1. +-commutative 50.2%

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

      2. fma-def 50.2%

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

      3. unpow2 50.2%

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

      4. *-commutative 50.2%

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

      5. times-frac 50.1%

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

      6. associate-*r/ 50.1%

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

      7. *-commutative 50.1%

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

      8. times-frac 50.3%

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

      9. unpow2 50.3%

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

      10. unpow2 50.3%

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

      11. times-frac 52.3%

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

      12. unpow2 52.3%

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

      13. associate-/l* 52.7%

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

      14. distribute-rgt-out 52.7%

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

      15. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   10. Taylor expanded in l around 0 59.1%

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

       1. unpow2 59.1%

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

       2. associate-*r/ 59.1%

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

       3. metadata-eval 59.1%

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

       4. *-commutative 59.1%

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

       5. associate-*r/ 59.1%

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

       6. metadata-eval 59.1%

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

       7. unpow2 59.1%

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

   12. Simplified 59.1%

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

   13. Taylor expanded in k around inf 59.1%

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

       1. unpow2 59.1%

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

       2. associate-*r* 59.3%

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

   15. Simplified 59.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+29}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\frac{2}{k}}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 12: 69.2% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k}}{t} \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e+29)
   (* (/ (/ (/ 2.0 k) k) t) (/ l (* k (/ k l))))
   (* (* l l) (/ -0.3333333333333333 (* k (* k t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e+29) {
		tmp = (((2.0 / k) / k) / t) * (l / (k * (k / l)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	}
	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 <= 1.95d+29) then
        tmp = (((2.0d0 / k) / k) / t) * (l / (k * (k / l)))
    else
        tmp = (l * l) * ((-0.3333333333333333d0) / (k * (k * t)))
    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 <= 1.95e+29) {
		tmp = (((2.0 / k) / k) / t) * (l / (k * (k / l)));
	} else {
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.95e+29:
		tmp = (((2.0 / k) / k) / t) * (l / (k * (k / l)))
	else:
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e+29)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / k) / t) * Float64(l / Float64(k * Float64(k / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e+29)
		tmp = (((2.0 / k) / k) / t) * (l / (k * (k / l)));
	else
		tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.95e+29], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999984e29

   1. Initial program 34.7%

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

      1. associate-*l* 34.7%

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

      2. associate-*l* 34.7%

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

      3. associate-/r* 34.7%

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

      4. associate-/r/ 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 35.3%

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

      7. +-commutative 35.3%

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

      8. associate--l+ 42.7%

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

      9. metadata-eval 42.7%

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

      10. +-rgt-identity 42.7%

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

      11. times-frac 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in t around 0 82.3%

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

      1. associate-/r* 82.4%

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

      2. unpow2 82.4%

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

      3. associate-/r* 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in k around 0 68.8%

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

      1. unpow2 68.8%

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

      2. unpow2 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in l around 0 68.8%

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

       1. unpow2 68.8%

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

       2. associate-/l* 75.8%

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

       3. unpow2 75.8%

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

       4. associate-*r/ 75.8%

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

   12. Simplified 75.8%

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

   if 1.94999999999999984e29 < k

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-/r* 36.1%

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

      4. associate-/r/ 34.6%

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

      5. *-commutative 34.6%

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

      6. times-frac 33.3%

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

      7. +-commutative 33.3%

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

      8. associate--l+ 46.9%

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

      9. metadata-eval 46.9%

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

      10. +-rgt-identity 46.9%

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

      11. times-frac 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in t around 0 81.1%

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

      1. associate-/r* 81.1%

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

      2. unpow2 81.1%

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

      3. associate-/r* 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in k around 0 50.2%

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

      1. +-commutative 50.2%

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

      2. fma-def 50.2%

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

      3. unpow2 50.2%

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

      4. *-commutative 50.2%

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

      5. times-frac 50.1%

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

      6. associate-*r/ 50.1%

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

      7. *-commutative 50.1%

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

      8. times-frac 50.3%

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

      9. unpow2 50.3%

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

      10. unpow2 50.3%

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

      11. times-frac 52.3%

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

      12. unpow2 52.3%

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

      13. associate-/l* 52.7%

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

      14. distribute-rgt-out 52.7%

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

      15. metadata-eval 52.7%

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

   9. Simplified 52.7%

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

   10. Taylor expanded in l around 0 59.1%

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

       1. unpow2 59.1%

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

       2. associate-*r/ 59.1%

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

       3. metadata-eval 59.1%

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

       4. *-commutative 59.1%

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

       5. associate-*r/ 59.1%

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

       6. metadata-eval 59.1%

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

       7. unpow2 59.1%

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

   12. Simplified 59.1%

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

   13. Taylor expanded in k around inf 59.1%

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

       1. unpow2 59.1%

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

       2. associate-*r* 59.3%

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

   15. Simplified 59.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k}}{t} \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 13: 70.0% accurate, 28.1× speedup

Math

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

FPCore

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

C

k = abs(k);
double code(double t, double l, double k) {
	return (2.0 * ((l / k) * (l / k))) / (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 = (2.0d0 * ((l / k) * (l / k))) / (k * (k * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. *-commutative 35.0%

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

   6. times-frac 34.8%

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

   7. +-commutative 34.8%

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

   8. associate--l+ 43.8%

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

   9. metadata-eval 43.8%

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

   10. +-rgt-identity 43.8%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 82.0%

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

   1. associate-/r* 82.0%

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

   2. unpow2 82.0%

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

   3. associate-/r* 82.4%

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

6. Simplified 82.4%

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

7. Taylor expanded in k around 0 66.5%

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

   1. unpow2 66.5%

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

   2. unpow2 66.5%

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

9. Simplified 66.5%

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

10. Taylor expanded in k around 0 66.5%

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

    1. *-commutative 66.5%

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

    2. unpow2 66.5%

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

12. Simplified 66.5%

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

    1. associate-*l/ 66.6%

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

    2. times-frac 71.7%

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

    3. *-commutative 71.7%

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

    4. associate-*l* 73.2%

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

14. Applied egg-rr 73.2%

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

15. Final simplification 73.2%

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

Alternative 14: 70.9% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

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) / (k * t))) / (k / (l / k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.0%

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

   4. associate-/r/ 35.0%

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

   5. *-commutative 35.0%

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

   6. times-frac 34.8%

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

   7. +-commutative 34.8%

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

   8. associate--l+ 43.8%

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

   9. metadata-eval 43.8%

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

   10. +-rgt-identity 43.8%

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

   11. times-frac 47.0%

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

3. Simplified 47.0%

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

4. Taylor expanded in t around 0 82.0%

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

   1. associate-/r* 82.0%

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

   2. unpow2 82.0%

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

   3. associate-/r* 82.4%

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

6. Simplified 82.4%

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

   1. *-commutative 82.4%

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

   2. clear-num 82.4%

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

   3. frac-times 82.5%

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

   4. *-un-lft-identity 82.5%

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

8. Applied egg-rr 82.5%

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

   1. associate-*r/ 85.2%

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

   2. associate-/l/ 88.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{k}}{t \cdot k}} \cdot \ell}{\frac{\tan k}{\ell} \cdot \sin k} \]

   3. associate-*l/ 87.2%

      \[\leadsto \frac{\frac{\frac{2}{k}}{t \cdot k} \cdot \ell}{\color{blue}{\frac{\tan k \cdot \sin k}{\ell}}} \]

10. Applied egg-rr 87.2%

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{k}}{t \cdot k} \cdot \ell}{\frac{\tan k \cdot \sin k}{\ell}}} \]

11. Taylor expanded in k around 0 71.9%

    \[\leadsto \frac{\frac{\frac{2}{k}}{t \cdot k} \cdot \ell}{\color{blue}{\frac{{k}^{2}}{\ell}}} \]
12. Step-by-step derivation

    1. unpow2 71.9%

       \[\leadsto \frac{\frac{\frac{2}{k}}{t \cdot k} \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\ell}} \]

    2. associate-/l* 73.4%

       \[\leadsto \frac{\frac{\frac{2}{k}}{t \cdot k} \cdot \ell}{\color{blue}{\frac{k}{\frac{\ell}{k}}}} \]

13. Simplified 73.4%

    \[\leadsto \frac{\frac{\frac{2}{k}}{t \cdot k} \cdot \ell}{\color{blue}{\frac{k}{\frac{\ell}{k}}}} \]

14. Final simplification 73.4%

    \[\leadsto \frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{k}{\frac{\ell}{k}}} \]

Alternative 15: 33.8% accurate, 38.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ -0.3333333333333333 (* k (* k t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return (l * l) * (-0.3333333333333333 / (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 * l) * ((-0.3333333333333333d0) / (k * (k * t)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (l * l) * (-0.3333333333333333 / (k * (k * t)));
}

Python

k = abs(k)
def code(t, l, k):
	return (l * l) * (-0.3333333333333333 / (k * (k * t)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.3333333333333333 / Float64(k * Float64(k * t))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (l * l) * (-0.3333333333333333 / (k * (k * t)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 35.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 35.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 35.0%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 35.0%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 34.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 34.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 43.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 43.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 43.8%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 47.0%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 47.0%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 82.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. associate-/r* 82.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   2. unpow2 82.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   3. associate-/r* 82.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{k}}{k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 82.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{k}}{k}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

7. Taylor expanded in k around 0 32.9%

   \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. +-commutative 32.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]

   2. fma-def 32.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]

   3. unpow2 32.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   4. *-commutative 32.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   5. times-frac 33.4%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   6. associate-*r/ 33.4%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \color{blue}{\frac{-2 \cdot \left({\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)}{{k}^{2} \cdot {t}^{2}}}\right) \]

   7. *-commutative 33.4%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2 \cdot \left({\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)}{\color{blue}{{t}^{2} \cdot {k}^{2}}}\right) \]

   8. times-frac 31.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \color{blue}{\frac{-2}{{t}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2}}}\right) \]

   9. unpow2 31.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{\color{blue}{t \cdot t}} \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2}}\right) \]

   10. unpow2 31.5%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{\color{blue}{k \cdot k}}\right) \]

   11. times-frac 35.0%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{k} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{k}\right)}\right) \]

   12. unpow2 35.0%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{k} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{k}\right)\right) \]

   13. associate-/l* 34.4%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\color{blue}{\frac{\ell}{\frac{k}{\ell}}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{k}\right)\right) \]

   14. distribute-rgt-out 34.4%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}{k}\right)\right) \]

   15. metadata-eval 34.4%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{t \cdot \color{blue}{0.16666666666666666}}{k}\right)\right) \]

9. Simplified 34.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{t \cdot 0.16666666666666666}{k}\right)\right)} \]

10. Taylor expanded in l around 0 47.7%

    \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. unpow2 47.7%

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    2. associate-*r/ 47.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    3. metadata-eval 47.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    4. *-commutative 47.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    5. associate-*r/ 48.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right) \]

    6. metadata-eval 48.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \frac{\color{blue}{0.3333333333333333}}{{k}^{2} \cdot t}\right) \]

    7. unpow2 48.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \frac{0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

12. Simplified 48.1%

    \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \frac{0.3333333333333333}{\left(k \cdot k\right) \cdot t}\right)} \]

13. Taylor expanded in k around inf 35.8%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.3333333333333333}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. unpow2 35.8%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    2. associate-*r* 36.0%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

15. Simplified 36.0%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}} \]

16. Final simplification 36.0%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

Alternative 16: 33.8% accurate, 38.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ (* (* l l) -0.3333333333333333) (* k (* k t))))

C

k = abs(k);
double code(double t, double l, double k) {
	return ((l * l) * -0.3333333333333333) / (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 * l) * (-0.3333333333333333d0)) / (k * (k * t))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return ((l * l) * -0.3333333333333333) / (k * (k * t));
}

Python

k = abs(k)
def code(t, l, k):
	return ((l * l) * -0.3333333333333333) / (k * (k * t))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(k * Float64(k * t)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = ((l * l) * -0.3333333333333333) / (k * (k * t));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 35.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 35.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 35.0%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 35.0%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 34.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 34.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 43.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 43.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 43.8%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 47.0%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 47.0%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 82.0%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. associate-/r* 82.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   2. unpow2 82.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   3. associate-/r* 82.4%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{k}}{k}}}{t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 82.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{k}}{k}}{t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

7. Taylor expanded in k around 0 32.9%

   \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. +-commutative 32.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]

   2. fma-def 32.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]

   3. unpow2 32.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   4. *-commutative 32.9%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   5. times-frac 33.4%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   6. associate-*r/ 33.4%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \color{blue}{\frac{-2 \cdot \left({\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)}{{k}^{2} \cdot {t}^{2}}}\right) \]

   7. *-commutative 33.4%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2 \cdot \left({\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)}{\color{blue}{{t}^{2} \cdot {k}^{2}}}\right) \]

   8. times-frac 31.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \color{blue}{\frac{-2}{{t}^{2}} \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2}}}\right) \]

   9. unpow2 31.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{\color{blue}{t \cdot t}} \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2}}\right) \]

   10. unpow2 31.5%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{\color{blue}{k \cdot k}}\right) \]

   11. times-frac 35.0%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{k} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{k}\right)}\right) \]

   12. unpow2 35.0%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{k} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{k}\right)\right) \]

   13. associate-/l* 34.4%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\color{blue}{\frac{\ell}{\frac{k}{\ell}}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{k}\right)\right) \]

   14. distribute-rgt-out 34.4%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}{k}\right)\right) \]

   15. metadata-eval 34.4%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{t \cdot \color{blue}{0.16666666666666666}}{k}\right)\right) \]

9. Simplified 34.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{-2}{t \cdot t} \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{t \cdot 0.16666666666666666}{k}\right)\right)} \]

10. Taylor expanded in l around 0 47.7%

    \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. unpow2 47.7%

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    2. associate-*r/ 47.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    3. metadata-eval 47.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    4. *-commutative 47.7%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

    5. associate-*r/ 48.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right) \]

    6. metadata-eval 48.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \frac{\color{blue}{0.3333333333333333}}{{k}^{2} \cdot t}\right) \]

    7. unpow2 48.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \frac{0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]

12. Simplified 48.1%

    \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \frac{0.3333333333333333}{\left(k \cdot k\right) \cdot t}\right)} \]

13. Taylor expanded in k around inf 35.8%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. unpow2 35.8%

       \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    2. associate-*r* 36.0%

       \[\leadsto -0.3333333333333333 \cdot \frac{{\ell}^{2}}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

    3. associate-*r/ 36.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{k \cdot \left(k \cdot t\right)}} \]

    4. unpow2 36.0%

       \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{k \cdot \left(k \cdot t\right)} \]

15. Simplified 36.0%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)}} \]

16. Final simplification 36.0%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

Reproduce

herbie shell --seed 2023228 
(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))))