Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 94.0%

Time: 32.4s

Alternatives: 9

Speedup: 28.1×

• 34.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{{\sin k}^{2} \cdot t}\right) \cdot \frac{\cos k}{k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.05e-35)
   (/ (* (cos k) (/ l (* (* k 0.5) (* t (* k (/ k l)))))) k)
   (* 2.0 (* (* (/ l k) (/ l (* (pow (sin k) 2.0) t))) (/ (cos k) k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.05e-35) {
		tmp = (cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	} else {
		tmp = 2.0 * (((l / k) * (l / (pow(sin(k), 2.0) * t))) * (cos(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 <= 1.05d-35) then
        tmp = (cos(k) * (l / ((k * 0.5d0) * (t * (k * (k / l)))))) / k
    else
        tmp = 2.0d0 * (((l / k) * (l / ((sin(k) ** 2.0d0) * t))) * (cos(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 <= 1.05e-35) {
		tmp = (Math.cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	} else {
		tmp = 2.0 * (((l / k) * (l / (Math.pow(Math.sin(k), 2.0) * t))) * (Math.cos(k) / k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.05e-35:
		tmp = (math.cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k
	else:
		tmp = 2.0 * (((l / k) * (l / (math.pow(math.sin(k), 2.0) * t))) * (math.cos(k) / k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.05e-35)
		tmp = Float64(Float64(cos(k) * Float64(l / Float64(Float64(k * 0.5) * Float64(t * Float64(k * Float64(k / l)))))) / k);
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64((sin(k) ^ 2.0) * t))) * Float64(cos(k) / k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.05e-35)
		tmp = (cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	else
		tmp = 2.0 * (((l / k) * (l / ((sin(k) ^ 2.0) * t))) * (cos(k) / k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.05e-35], N[(N[(N[Cos[k], $MachinePrecision] * N[(l / N[(N[(k * 0.5), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e-35

   1. Initial program 44.4%

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

      1. associate-/r* 44.4%

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

      2. *-commutative 44.4%

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

      3. associate-/r* 48.4%

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

      4. associate-*r/ 48.5%

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

      5. associate-/l* 48.4%

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

      6. +-commutative 48.4%

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

      7. unpow2 48.4%

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

      8. sqr-neg 48.4%

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

      9. distribute-frac-neg 48.4%

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

      10. distribute-frac-neg 48.4%

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

      11. unpow2 48.4%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around inf 79.2%

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

      1. *-commutative 79.2%

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

      2. times-frac 79.5%

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

      3. associate-*l* 79.5%

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

      4. unpow2 79.5%

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

      5. *-commutative 79.5%

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

      6. associate-/r* 75.6%

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

      7. unpow2 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in k around inf 79.2%

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

      1. *-commutative 79.2%

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

      2. times-frac 79.5%

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

      3. unpow2 79.5%

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

      4. unpow2 79.5%

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

      5. *-commutative 79.5%

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

      6. associate-*l* 79.5%

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

      7. *-commutative 79.5%

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

      8. associate-/r* 79.5%

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

      9. associate-*l/ 80.5%

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

      10. associate-*r/ 80.5%

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

      11. *-commutative 80.5%

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

      12. associate-/l* 80.5%

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

   9. Simplified 89.3%

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

       1. associate-*l/ 89.3%

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

       2. associate-/l/ 91.4%

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

       3. div-inv 91.4%

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

       4. metadata-eval 91.4%

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

       5. associate-/r/ 94.4%

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

   11. Applied egg-rr 94.4%

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

   12. Taylor expanded in k around 0 85.4%

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

       1. unpow2 85.4%

          \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right)}}{k} \]

       2. associate-*r/ 87.4%

          \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

   14. Simplified 87.4%

       \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

   if 1.05e-35 < k

   1. Initial program 27.3%

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

      1. associate-/r* 27.3%

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

      2. *-commutative 27.3%

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

      3. associate-/r* 30.2%

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

      4. associate-*r/ 30.2%

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

      5. associate-/l* 30.2%

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

      6. +-commutative 30.2%

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

      7. unpow2 30.2%

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

      8. sqr-neg 30.2%

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

      9. distribute-frac-neg 30.2%

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

      10. distribute-frac-neg 30.2%

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

      11. unpow2 30.2%

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

      12. associate--l+ 36.4%

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

      13. metadata-eval 36.4%

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

      14. +-rgt-identity 36.4%

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

      15. unpow2 36.4%

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

      16. distribute-frac-neg 36.4%

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

   3. Simplified 36.4%

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

   4. Taylor expanded in k around inf 72.4%

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

      1. *-commutative 72.4%

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

      2. unpow2 72.4%

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

      3. unpow2 72.4%

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

      4. *-commutative 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in k around inf 72.4%

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

      1. unpow2 72.4%

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

      2. *-commutative 72.4%

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

      3. associate-*l* 77.1%

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

      4. *-commutative 77.1%

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

   9. Simplified 77.1%

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

       1. times-frac 81.9%

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

   11. Applied egg-rr 81.9%

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

   12. Taylor expanded in l around 0 72.4%

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

       1. *-commutative 72.4%

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

       2. unpow2 72.4%

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

       3. associate-*r* 77.1%

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

       4. *-commutative 77.1%

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

       5. times-frac 80.8%

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

       6. unpow2 80.8%

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

       7. times-frac 96.1%

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

   14. Simplified 96.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{{\sin k}^{2} \cdot t}\right) \cdot \frac{\cos k}{k}\right)\\ \end{array} \]

Alternative 2: 83.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 2.2e-129)
   (/ (* (cos k) (/ l (* (* k 0.5) (* t (* k (/ k l)))))) k)
   (* 2.0 (* (* l (/ l k)) (/ (cos k) (* k (* (pow (sin k) 2.0) t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.2e-129) {
		tmp = (cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	} else {
		tmp = 2.0 * ((l * (l / k)) * (cos(k) / (k * (pow(sin(k), 2.0) * 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 (l <= 2.2d-129) then
        tmp = (cos(k) * (l / ((k * 0.5d0) * (t * (k * (k / l)))))) / k
    else
        tmp = 2.0d0 * ((l * (l / k)) * (cos(k) / (k * ((sin(k) ** 2.0d0) * 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 (l <= 2.2e-129) {
		tmp = (Math.cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	} else {
		tmp = 2.0 * ((l * (l / k)) * (Math.cos(k) / (k * (Math.pow(Math.sin(k), 2.0) * t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if l <= 2.2e-129:
		tmp = (math.cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k
	else:
		tmp = 2.0 * ((l * (l / k)) * (math.cos(k) / (k * (math.pow(math.sin(k), 2.0) * t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (l <= 2.2e-129)
		tmp = Float64(Float64(cos(k) * Float64(l / Float64(Float64(k * 0.5) * Float64(t * Float64(k * Float64(k / l)))))) / k);
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / k)) * Float64(cos(k) / Float64(k * Float64((sin(k) ^ 2.0) * t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 2.2e-129)
		tmp = (cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	else
		tmp = 2.0 * ((l * (l / k)) * (cos(k) / (k * ((sin(k) ^ 2.0) * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 2.2e-129], N[(N[(N[Cos[k], $MachinePrecision] * N[(l / N[(N[(k * 0.5), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.20000000000000003e-129

   1. Initial program 36.9%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-/r* 41.6%

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

      4. associate-*r/ 41.6%

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

      5. associate-/l* 41.6%

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

      6. +-commutative 41.6%

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

      7. unpow2 41.6%

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

      8. sqr-neg 41.6%

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

      9. distribute-frac-neg 41.6%

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

      10. distribute-frac-neg 41.6%

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

      11. unpow2 41.6%

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

      12. associate--l+ 48.2%

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

      13. metadata-eval 48.2%

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

      14. +-rgt-identity 48.2%

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

      15. unpow2 48.2%

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

      16. distribute-frac-neg 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. times-frac 76.9%

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

      3. associate-*l* 76.9%

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

      4. unpow2 76.9%

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

      5. *-commutative 76.9%

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

      6. associate-/r* 73.8%

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

      7. unpow2 73.8%

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

   6. Simplified 73.8%

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

   7. Taylor expanded in k around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. times-frac 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. *-commutative 76.9%

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

      6. associate-*l* 76.9%

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

      7. *-commutative 76.9%

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

      8. associate-/r* 77.2%

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

      9. associate-*l/ 80.1%

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

      10. associate-*r/ 80.1%

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

      11. *-commutative 80.1%

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

      12. associate-/l* 80.1%

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

   9. Simplified 91.4%

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

       1. associate-*l/ 91.4%

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

       2. associate-/l/ 93.1%

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

       3. div-inv 93.1%

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

       4. metadata-eval 93.1%

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

       5. associate-/r/ 96.3%

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

   11. Applied egg-rr 96.3%

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

   12. Taylor expanded in k around 0 81.7%

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

       1. unpow2 81.7%

          \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right)}}{k} \]

       2. associate-*r/ 83.8%

          \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

   14. Simplified 83.8%

       \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

   if 2.20000000000000003e-129 < l

   1. Initial program 42.3%

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

      1. associate-/r* 42.3%

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

      2. *-commutative 42.3%

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

      3. associate-/r* 44.3%

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

      4. associate-*r/ 44.3%

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

      5. associate-/l* 44.3%

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

      6. +-commutative 44.3%

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

      7. unpow2 44.3%

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

      8. sqr-neg 44.3%

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

      9. distribute-frac-neg 44.3%

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

      10. distribute-frac-neg 44.3%

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

      11. unpow2 44.3%

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

      12. associate--l+ 45.4%

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

      13. metadata-eval 45.4%

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

      14. +-rgt-identity 45.4%

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

      15. unpow2 45.4%

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

      16. distribute-frac-neg 45.4%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in k around inf 76.1%

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

      1. *-commutative 76.1%

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

      2. unpow2 76.1%

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

      3. unpow2 76.1%

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

      4. *-commutative 76.1%

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

   6. Simplified 76.1%

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

   7. Taylor expanded in k around inf 76.1%

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

      1. unpow2 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*l* 78.3%

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

      4. *-commutative 78.3%

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

   9. Simplified 78.3%

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

       1. times-frac 82.4%

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

   11. Applied egg-rr 82.4%

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

   12. Taylor expanded in l around 0 82.4%

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

       1. unpow2 82.4%

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

       2. associate-*r/ 92.4%

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

   14. Simplified 92.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{k \cdot \left({\sin k}^{2} \cdot t\right)}\right)\\ \end{array} \]

Alternative 3: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot t\right)}}{k} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ (* (cos k) (/ l (* (* k 0.5) (* (/ (pow (sin k) 2.0) l) t)))) k))

C

k = abs(k);
double code(double t, double l, double k) {
	return (cos(k) * (l / ((k * 0.5) * ((pow(sin(k), 2.0) / l) * t)))) / 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 = (cos(k) * (l / ((k * 0.5d0) * (((sin(k) ** 2.0d0) / l) * t)))) / k
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (Math.cos(k) * (l / ((k * 0.5) * ((Math.pow(Math.sin(k), 2.0) / l) * t)))) / k;
}

Python

k = abs(k)
def code(t, l, k):
	return (math.cos(k) * (l / ((k * 0.5) * ((math.pow(math.sin(k), 2.0) / l) * t)))) / k

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(cos(k) * Float64(l / Float64(Float64(k * 0.5) * Float64(Float64((sin(k) ^ 2.0) / l) * t)))) / k)
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (cos(k) * (l / ((k * 0.5) * (((sin(k) ^ 2.0) / l) * t)))) / k;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[Cos[k], $MachinePrecision] * N[(l / N[(N[(k * 0.5), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

   1. associate-/r* 38.9%

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

   2. *-commutative 38.9%

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

   3. associate-/r* 42.6%

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

   4. associate-*r/ 42.6%

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

   5. associate-/l* 42.6%

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

   6. +-commutative 42.6%

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

   7. unpow2 42.6%

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

   8. sqr-neg 42.6%

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

   9. distribute-frac-neg 42.6%

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

   10. distribute-frac-neg 42.6%

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

   11. unpow2 42.6%

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

   12. associate--l+ 47.2%

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

   13. metadata-eval 47.2%

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

   14. +-rgt-identity 47.2%

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

   15. unpow2 47.2%

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

   16. distribute-frac-neg 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. associate-*l* 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-/r* 73.1%

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

   7. unpow2 73.1%

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

6. Simplified 73.1%

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

7. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. unpow2 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-*l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-/r* 76.6%

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

   9. associate-*l/ 79.9%

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

   10. associate-*r/ 79.9%

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

   11. *-commutative 79.9%

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

   12. associate-/l* 79.9%

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

9. Simplified 88.9%

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

    1. associate-*l/ 88.9%

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

    2. associate-/l/ 93.0%

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

    3. div-inv 93.0%

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

    4. metadata-eval 93.0%

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

    5. associate-/r/ 95.0%

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

11. Applied egg-rr 95.0%

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

12. Final simplification 95.0%

    \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot t\right)}}{k} \]

Alternative 4: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \frac{\mathsf{fma}\left(\ell \cdot \frac{\ell}{t}, 0.3333333333333333, \frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}\right)}{\frac{k}{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.5e-42)
   (/ (* (cos k) (/ l (* (* k 0.5) (* t (* k (/ k l)))))) k)
   (*
    (/ (cos k) k)
    (/
     (fma (* l (/ l t)) 0.3333333333333333 (/ l (/ (* k (* k t)) l)))
     (/ k 2.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.5e-42) {
		tmp = (cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
	} else {
		tmp = (cos(k) / k) * (fma((l * (l / t)), 0.3333333333333333, (l / ((k * (k * t)) / l))) / (k / 2.0));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.5e-42)
		tmp = Float64(Float64(cos(k) * Float64(l / Float64(Float64(k * 0.5) * Float64(t * Float64(k * Float64(k / l)))))) / k);
	else
		tmp = Float64(Float64(cos(k) / k) * Float64(fma(Float64(l * Float64(l / t)), 0.3333333333333333, Float64(l / Float64(Float64(k * Float64(k * t)) / l))) / Float64(k / 2.0)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.5e-42], N[(N[(N[Cos[k], $MachinePrecision] * N[(l / N[(N[(k * 0.5), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(l / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5000000000000002e-42

   1. Initial program 44.1%

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

      1. associate-/r* 44.1%

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

      2. *-commutative 44.1%

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

      3. associate-/r* 48.1%

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

      4. associate-*r/ 48.2%

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

      5. associate-/l* 48.1%

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

      6. +-commutative 48.1%

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

      7. unpow2 48.1%

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

      8. sqr-neg 48.1%

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

      9. distribute-frac-neg 48.1%

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

      10. distribute-frac-neg 48.1%

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

      11. unpow2 48.1%

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

      12. associate--l+ 52.0%

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

      13. metadata-eval 52.0%

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

      14. +-rgt-identity 52.0%

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

      15. unpow2 52.0%

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

      16. distribute-frac-neg 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in k around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. times-frac 79.4%

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

      3. associate-*l* 79.4%

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

      4. unpow2 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-/r* 75.4%

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

      7. unpow2 75.4%

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

   6. Simplified 75.4%

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

   7. Taylor expanded in k around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. times-frac 79.4%

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

      3. unpow2 79.4%

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

      4. unpow2 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l* 79.4%

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

      7. *-commutative 79.4%

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

      8. associate-/r* 79.4%

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

      9. associate-*l/ 80.4%

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

      10. associate-*r/ 80.4%

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

      11. *-commutative 80.4%

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

      12. associate-/l* 80.4%

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

   9. Simplified 89.2%

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

       1. associate-*l/ 89.2%

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

       2. associate-/l/ 91.4%

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

       3. div-inv 91.4%

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

       4. metadata-eval 91.4%

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

       5. associate-/r/ 94.4%

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

   11. Applied egg-rr 94.4%

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

   12. Taylor expanded in k around 0 85.3%

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

       1. unpow2 85.3%

          \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right)}}{k} \]

       2. associate-*r/ 87.3%

          \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

   14. Simplified 87.3%

       \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

   if 3.5000000000000002e-42 < k

   1. Initial program 28.2%

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

      1. associate-/r* 28.2%

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

      2. *-commutative 28.2%

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

      3. associate-/r* 31.0%

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

      4. associate-*r/ 31.0%

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

      5. associate-/l* 31.0%

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

      6. +-commutative 31.0%

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

      7. unpow2 31.0%

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

      8. sqr-neg 31.0%

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

      9. distribute-frac-neg 31.0%

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

      10. distribute-frac-neg 31.0%

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

      11. unpow2 31.0%

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

      12. associate--l+ 37.2%

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

      13. metadata-eval 37.2%

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

      14. +-rgt-identity 37.2%

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

      15. unpow2 37.2%

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

      16. distribute-frac-neg 37.2%

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

   3. Simplified 37.2%

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

   4. Taylor expanded in k around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 68.4%

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

      3. associate-*l* 68.4%

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

      4. unpow2 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-/r* 68.4%

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

      7. unpow2 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in k around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. times-frac 68.4%

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

      3. unpow2 68.4%

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

      4. unpow2 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-*l* 68.4%

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

      7. *-commutative 68.4%

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

      8. associate-/r* 70.6%

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

      9. associate-*l/ 78.9%

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

      10. associate-*r/ 78.9%

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

      11. *-commutative 78.9%

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

      12. associate-/l* 78.9%

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

   9. Simplified 88.2%

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

   10. Taylor expanded in k around 0 61.5%

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

       1. +-commutative 61.5%

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

       2. *-commutative 61.5%

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

       3. fma-def 61.5%

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

       4. unpow2 61.5%

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

       5. associate-*r/ 61.6%

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

       6. unpow2 61.6%

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

       7. associate-/l* 65.5%

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

       8. unpow2 65.5%

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

       9. associate-*l* 65.5%

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

   12. Simplified 65.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{k} \cdot \frac{\mathsf{fma}\left(\ell \cdot \frac{\ell}{t}, 0.3333333333333333, \frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}\right)}{\frac{k}{2}}\\ \end{array} \]

Alternative 5: 70.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := t \cdot \left(k \cdot k\right)\\ \mathbf{if}\;\ell \leq 1.82 \cdot 10^{+220}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\frac{t_1}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (* k k))))
   (if (<= l 1.82e+220)
     (/ (* 2.0 (/ l (/ t_1 l))) (* k k))
     (* 2.0 (/ (* (cos k) (* l l)) (* (* k k) t_1))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = t * (k * k);
	double tmp;
	if (l <= 1.82e+220) {
		tmp = (2.0 * (l / (t_1 / l))) / (k * k);
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / ((k * k) * t_1));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (k * k)
    if (l <= 1.82d+220) then
        tmp = (2.0d0 * (l / (t_1 / l))) / (k * k)
    else
        tmp = 2.0d0 * ((cos(k) * (l * l)) / ((k * k) * t_1))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = t * (k * k);
	double tmp;
	if (l <= 1.82e+220) {
		tmp = (2.0 * (l / (t_1 / l))) / (k * k);
	} else {
		tmp = 2.0 * ((Math.cos(k) * (l * l)) / ((k * k) * t_1));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = t * (k * k)
	tmp = 0
	if l <= 1.82e+220:
		tmp = (2.0 * (l / (t_1 / l))) / (k * k)
	else:
		tmp = 2.0 * ((math.cos(k) * (l * l)) / ((k * k) * t_1))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(t * Float64(k * k))
	tmp = 0.0
	if (l <= 1.82e+220)
		tmp = Float64(Float64(2.0 * Float64(l / Float64(t_1 / l))) / Float64(k * k));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(l * l)) / Float64(Float64(k * k) * t_1)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = t * (k * k);
	tmp = 0.0;
	if (l <= 1.82e+220)
		tmp = (2.0 * (l / (t_1 / l))) / (k * k);
	else
		tmp = 2.0 * ((cos(k) * (l * l)) / ((k * k) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.82e+220], N[(N[(2.0 * N[(l / N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.81999999999999996e220

   1. Initial program 39.1%

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

      1. associate-/r* 39.1%

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

      2. *-commutative 39.1%

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

      3. associate-/r* 42.9%

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

      4. associate-*r/ 42.9%

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

      5. associate-/l* 42.9%

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

      6. +-commutative 42.9%

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

      7. unpow2 42.9%

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

      8. sqr-neg 42.9%

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

      9. distribute-frac-neg 42.9%

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

      10. distribute-frac-neg 42.9%

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

      11. unpow2 42.9%

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

      12. associate--l+ 47.8%

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

      13. metadata-eval 47.8%

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

      14. +-rgt-identity 47.8%

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

      15. unpow2 47.8%

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

      16. distribute-frac-neg 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. times-frac 76.8%

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

      3. associate-*l* 76.8%

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

      4. unpow2 76.8%

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

      5. *-commutative 76.8%

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

      6. associate-/r* 73.9%

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

      7. unpow2 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in k around 0 66.5%

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

      1. unpow2 66.5%

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

   9. Simplified 66.5%

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

   10. Taylor expanded in k around 0 68.7%

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

       1. unpow2 68.7%

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

       2. associate-/l* 75.2%

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

       3. unpow2 75.2%

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

       4. associate-*l* 75.1%

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

   12. Simplified 75.1%

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

       1. associate-*l/ 75.3%

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

       2. *-un-lft-identity 75.3%

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

       3. *-commutative 75.3%

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

       4. associate-*r* 75.3%

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

   14. Applied egg-rr 75.3%

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

   if 1.81999999999999996e220 < l

   1. Initial program 37.1%

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

      1. associate-/r* 37.1%

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

      2. *-commutative 37.1%

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

      3. associate-/r* 38.8%

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

      4. associate-*r/ 38.8%

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

      5. associate-/l* 38.8%

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

      6. +-commutative 38.8%

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

      7. unpow2 38.8%

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

      8. sqr-neg 38.8%

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

      9. distribute-frac-neg 38.8%

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

      10. distribute-frac-neg 38.8%

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

      11. unpow2 38.8%

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

      12. associate--l+ 39.2%

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

      13. metadata-eval 39.2%

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

      14. +-rgt-identity 39.2%

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

      15. unpow2 39.2%

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

      16. distribute-frac-neg 39.2%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in k around inf 63.2%

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

      1. *-commutative 63.2%

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

      2. unpow2 63.2%

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

      3. unpow2 63.2%

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

      4. *-commutative 63.2%

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

   6. Simplified 63.2%

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

   7. Taylor expanded in k around 0 63.2%

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

      1. unpow2 63.2%

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

   9. Simplified 63.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.82 \cdot 10^{+220}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]

Alternative 6: 73.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

   1. associate-/r* 38.9%

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

   2. *-commutative 38.9%

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

   3. associate-/r* 42.6%

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

   4. associate-*r/ 42.6%

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

   5. associate-/l* 42.6%

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

   6. +-commutative 42.6%

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

   7. unpow2 42.6%

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

   8. sqr-neg 42.6%

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

   9. distribute-frac-neg 42.6%

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

   10. distribute-frac-neg 42.6%

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

   11. unpow2 42.6%

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

   12. associate--l+ 47.2%

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

   13. metadata-eval 47.2%

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

   14. +-rgt-identity 47.2%

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

   15. unpow2 47.2%

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

   16. distribute-frac-neg 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. associate-*l* 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-/r* 73.1%

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

   7. unpow2 73.1%

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

6. Simplified 73.1%

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

7. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. unpow2 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-*l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-/r* 76.6%

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

   9. associate-*l/ 79.9%

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

   10. associate-*r/ 79.9%

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

   11. *-commutative 79.9%

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

   12. associate-/l* 79.9%

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

9. Simplified 88.9%

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

10. Taylor expanded in k around 0 76.6%

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

    1. unpow2 76.6%

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

    2. associate-*l* 77.2%

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

12. Simplified 77.2%

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

13. Taylor expanded in l around 0 70.4%

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

    1. unpow2 70.4%

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

    2. times-frac 75.9%

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

    3. unpow2 75.9%

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

    4. associate-/r* 76.4%

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

15. Simplified 76.4%

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

16. Final simplification 76.4%

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

Alternative 7: 73.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

   1. associate-/r* 38.9%

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

   2. *-commutative 38.9%

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

   3. associate-/r* 42.6%

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

   4. associate-*r/ 42.6%

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

   5. associate-/l* 42.6%

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

   6. +-commutative 42.6%

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

   7. unpow2 42.6%

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

   8. sqr-neg 42.6%

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

   9. distribute-frac-neg 42.6%

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

   10. distribute-frac-neg 42.6%

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

   11. unpow2 42.6%

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

   12. associate--l+ 47.2%

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

   13. metadata-eval 47.2%

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

   14. +-rgt-identity 47.2%

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

   15. unpow2 47.2%

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

   16. distribute-frac-neg 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. associate-*l* 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-/r* 73.1%

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

   7. unpow2 73.1%

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

6. Simplified 73.1%

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

7. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. unpow2 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-*l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-/r* 76.6%

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

   9. associate-*l/ 79.9%

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

   10. associate-*r/ 79.9%

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

   11. *-commutative 79.9%

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

   12. associate-/l* 79.9%

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

9. Simplified 88.9%

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

10. Taylor expanded in k around 0 76.6%

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

    1. unpow2 76.6%

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

    2. associate-*l* 77.2%

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

12. Simplified 77.2%

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

13. Final simplification 77.2%

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

Alternative 8: 74.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ (* (cos k) (/ l (* (* k 0.5) (* t (* k (/ k l)))))) k))

C

k = abs(k);
double code(double t, double l, double k) {
	return (cos(k) * (l / ((k * 0.5) * (t * (k * (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 = (cos(k) * (l / ((k * 0.5d0) * (t * (k * (k / l)))))) / k
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return (Math.cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
}

Python

k = abs(k)
def code(t, l, k):
	return (math.cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(cos(k) * Float64(l / Float64(Float64(k * 0.5) * Float64(t * Float64(k * Float64(k / l)))))) / k)
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = (cos(k) * (l / ((k * 0.5) * (t * (k * (k / l)))))) / k;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[Cos[k], $MachinePrecision] * N[(l / N[(N[(k * 0.5), $MachinePrecision] * N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]

Derivation

1. Initial program 38.9%

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

   1. associate-/r* 38.9%

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

   2. *-commutative 38.9%

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

   3. associate-/r* 42.6%

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

   4. associate-*r/ 42.6%

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

   5. associate-/l* 42.6%

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

   6. +-commutative 42.6%

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

   7. unpow2 42.6%

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

   8. sqr-neg 42.6%

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

   9. distribute-frac-neg 42.6%

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

   10. distribute-frac-neg 42.6%

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

   11. unpow2 42.6%

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

   12. associate--l+ 47.2%

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

   13. metadata-eval 47.2%

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

   14. +-rgt-identity 47.2%

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

   15. unpow2 47.2%

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

   16. distribute-frac-neg 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. associate-*l* 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-/r* 73.1%

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

   7. unpow2 73.1%

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

6. Simplified 73.1%

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

7. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. unpow2 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-*l* 75.9%

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

   7. *-commutative 75.9%

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

   8. associate-/r* 76.6%

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

   9. associate-*l/ 79.9%

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

   10. associate-*r/ 79.9%

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

   11. *-commutative 79.9%

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

   12. associate-/l* 79.9%

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

9. Simplified 88.9%

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

    1. associate-*l/ 88.9%

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

    2. associate-/l/ 93.0%

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

    3. div-inv 93.0%

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

    4. metadata-eval 93.0%

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

    5. associate-/r/ 95.0%

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

11. Applied egg-rr 95.0%

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

12. Taylor expanded in k around 0 77.2%

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

    1. unpow2 77.2%

       \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right)}}{k} \]

    2. associate-*r/ 78.6%

       \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

14. Simplified 78.6%

    \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right)}}{k} \]

15. Final simplification 78.6%

    \[\leadsto \frac{\cos k \cdot \frac{\ell}{\left(k \cdot 0.5\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}}{k} \]

Alternative 9: 70.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.9%

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

   1. associate-/r* 38.9%

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

   2. *-commutative 38.9%

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

   3. associate-/r* 42.6%

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

   4. associate-*r/ 42.6%

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

   5. associate-/l* 42.6%

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

   6. +-commutative 42.6%

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

   7. unpow2 42.6%

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

   8. sqr-neg 42.6%

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

   9. distribute-frac-neg 42.6%

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

   10. distribute-frac-neg 42.6%

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

   11. unpow2 42.6%

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

   12. associate--l+ 47.2%

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

   13. metadata-eval 47.2%

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

   14. +-rgt-identity 47.2%

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

   15. unpow2 47.2%

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

   16. distribute-frac-neg 47.2%

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

3. Simplified 47.2%

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

4. Taylor expanded in k around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. times-frac 75.9%

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

   3. associate-*l* 75.9%

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

   4. unpow2 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-/r* 73.1%

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

   7. unpow2 73.1%

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

6. Simplified 73.1%

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

7. Taylor expanded in k around 0 65.1%

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

   1. unpow2 65.1%

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

9. Simplified 65.1%

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

10. Taylor expanded in k around 0 67.1%

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

    1. unpow2 67.1%

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

    2. associate-/l* 73.3%

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

    3. unpow2 73.3%

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

    4. associate-*l* 73.3%

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

12. Simplified 73.3%

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

    1. associate-*l/ 73.4%

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

    2. *-un-lft-identity 73.4%

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

    3. *-commutative 73.4%

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

    4. associate-*r* 73.4%

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

14. Applied egg-rr 73.4%

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

15. Final simplification 73.4%

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

Reproduce

herbie shell --seed 2023274 
(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))))