Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.8% → 90.8%

Time: 21.9s

Alternatives: 10

Speedup: 2.0×

• 35.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.8% 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: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-/r* 38.5%

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

   3. sub-neg 38.5%

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

   4. distribute-rgt-in 34.5%

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

   5. unpow2 34.5%

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

   6. times-frac 25.5%

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

   7. sqr-neg 25.5%

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

   8. times-frac 34.5%

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

   9. unpow2 34.5%

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

   10. distribute-rgt-in 38.5%

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

   11. +-commutative 38.5%

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

   12. associate-+l+ 43.5%

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

3. Simplified 43.5%

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

5. Taylor expanded in t around 0 76.3%

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

   1. times-frac 78.4%

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

7. Simplified 78.4%

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

   1. add-sqr-sqrt 78.4%

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

   2. pow2 78.4%

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

   3. div-inv 78.4%

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

   4. sqrt-prod 78.4%

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

   5. unpow2 78.4%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 85.1%

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

   8. pow-flip 85.1%

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

   9. metadata-eval 85.1%

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

9. Applied egg-rr 85.1%

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

10. Taylor expanded in k around 0 95.3%

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

    1. associate-*r/ 95.7%

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

    2. un-div-inv 95.8%

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

12. Applied egg-rr 95.8%

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

13. Final simplification 95.8%

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

Alternative 2: 48.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00016:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.00016)
   (pow (* (/ l (/ (pow k 2.0) (sqrt 2.0))) (sqrt (/ 1.0 t))) 2.0)
   (*
    2.0
    (*
     (pow (* l (/ 1.0 k)) 2.0)
     (/ (cos k) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.00016) {
		tmp = pow(((l / (pow(k, 2.0) / sqrt(2.0))) * sqrt((1.0 / t))), 2.0);
	} else {
		tmp = 2.0 * (pow((l * (1.0 / k)), 2.0) * (cos(k) / (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.00016d0) then
        tmp = ((l / ((k ** 2.0d0) / sqrt(2.0d0))) * sqrt((1.0d0 / t))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l * (1.0d0 / k)) ** 2.0d0) * (cos(k) / (t * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.00016) {
		tmp = Math.pow(((l / (Math.pow(k, 2.0) / Math.sqrt(2.0))) * Math.sqrt((1.0 / t))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l * (1.0 / k)), 2.0) * (Math.cos(k) / (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.00016:
		tmp = math.pow(((l / (math.pow(k, 2.0) / math.sqrt(2.0))) * math.sqrt((1.0 / t))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l * (1.0 / k)), 2.0) * (math.cos(k) / (t * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.00016)
		tmp = Float64(Float64(l / Float64((k ^ 2.0) / sqrt(2.0))) * sqrt(Float64(1.0 / t))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l * Float64(1.0 / k)) ^ 2.0) * Float64(cos(k) / Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.00016)
		tmp = ((l / ((k ^ 2.0) / sqrt(2.0))) * sqrt((1.0 / t))) ^ 2.0;
	else
		tmp = 2.0 * (((l * (1.0 / k)) ^ 2.0) * (cos(k) / (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.00016], N[Power[N[(N[(l / N[(N[Power[k, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.60000000000000013e-4

   1. Initial program 44.6%

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

      1. associate-*l* 44.6%

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

      2. associate-/r* 44.6%

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

      3. sub-neg 44.6%

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

      4. distribute-rgt-in 39.5%

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

      5. unpow2 39.5%

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

      6. times-frac 27.7%

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

      7. sqr-neg 27.7%

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

      8. times-frac 39.5%

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

      9. unpow2 39.5%

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

      10. distribute-rgt-in 44.6%

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

      11. +-commutative 44.6%

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

      12. associate-+l+ 49.5%

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

   3. Simplified 49.5%

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

   6. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot \frac{\sqrt{2}}{\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}} \]
   7. Step-by-step derivation

      1. unpow2 19.6%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt{2}}{\frac{k}{t} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}\right)}^{2}} \]

      2. associate-/r* 19.6%

         \[\leadsto {\color{blue}{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}}^{2} \]

   8. Simplified 19.6%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}} \]

   9. Taylor expanded in k around 0 26.7%

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

       1. associate-/l* 26.7%

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

   11. Simplified 26.7%

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

   if 1.60000000000000013e-4 < k

   1. Initial program 24.5%

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

      1. associate-*l* 24.5%

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

      2. associate-/r* 24.5%

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

      3. sub-neg 24.5%

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

      4. distribute-rgt-in 23.1%

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

      5. unpow2 23.1%

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

      6. times-frac 20.5%

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

      7. sqr-neg 20.5%

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

      8. times-frac 23.1%

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

      9. unpow2 23.1%

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

      10. distribute-rgt-in 24.5%

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

      11. +-commutative 24.5%

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

      12. associate-+l+ 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in t around 0 71.0%

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

      1. times-frac 68.4%

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

   7. Simplified 68.4%

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

      1. add-sqr-sqrt 68.4%

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

      2. pow2 68.4%

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

      3. div-inv 68.4%

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

      4. sqrt-prod 68.4%

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

      5. unpow2 68.4%

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

      6. sqrt-prod 44.0%

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

      7. add-sqr-sqrt 73.6%

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

      8. pow-flip 73.6%

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

      9. metadata-eval 73.6%

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

   9. Applied egg-rr 73.6%

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

   10. Taylor expanded in k around 0 93.2%

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

       1. unpow2 93.2%

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

       2. sin-mult 92.9%

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

   12. Applied egg-rr 92.9%

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

       1. div-sub 92.9%

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

       2. +-inverses 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

       3. cos-0 92.9%

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

       4. metadata-eval 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

       5. count-2 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}\right) \]

       6. *-commutative 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}\right) \]

   14. Simplified 92.9%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00016:\\ \;\;\;\;{\left(\frac{\ell}{\frac{{k}^{2}}{\sqrt{2}}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-/r* 38.5%

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

   3. sub-neg 38.5%

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

   4. distribute-rgt-in 34.5%

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

   5. unpow2 34.5%

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

   6. times-frac 25.5%

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

   7. sqr-neg 25.5%

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

   8. times-frac 34.5%

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

   9. unpow2 34.5%

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

   10. distribute-rgt-in 38.5%

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

   11. +-commutative 38.5%

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

   12. associate-+l+ 43.5%

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

3. Simplified 43.5%

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

5. Taylor expanded in t around 0 76.3%

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

   1. times-frac 78.4%

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

7. Simplified 78.4%

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

   1. add-sqr-sqrt 78.4%

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

   2. pow2 78.4%

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

   3. div-inv 78.4%

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

   4. sqrt-prod 78.4%

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

   5. unpow2 78.4%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 85.1%

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

   8. pow-flip 85.1%

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

   9. metadata-eval 85.1%

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

9. Applied egg-rr 85.1%

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

10. Taylor expanded in k around 0 95.3%

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

11. Taylor expanded in l around 0 76.3%

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

    1. times-frac 78.4%

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

    2. unpow2 78.4%

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

    3. unpow2 78.4%

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

    4. times-frac 95.4%

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

    5. unpow2 95.4%

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

    6. *-commutative 95.4%

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

    7. associate-/l/ 95.4%

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

13. Simplified 95.4%

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

14. Final simplification 95.4%

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

Alternative 4: 70.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\ell \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{if}\;k \leq 0.0136:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{t \cdot {k}^{2}}\right) + 0.16666666666666666 \cdot \frac{-1}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (* l (/ 1.0 k)) 2.0)))
   (if (<= k 0.0136)
     (*
      2.0
      (*
       t_1
       (+
        (+
         (* -0.058333333333333334 (/ (pow k 2.0) t))
         (/ 1.0 (* t (pow k 2.0))))
        (* 0.16666666666666666 (/ -1.0 t)))))
     (* 2.0 (* t_1 (/ (cos k) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((l * (1.0 / k)), 2.0);
	double tmp;
	if (k <= 0.0136) {
		tmp = 2.0 * (t_1 * (((-0.058333333333333334 * (pow(k, 2.0) / t)) + (1.0 / (t * pow(k, 2.0)))) + (0.16666666666666666 * (-1.0 / t))));
	} else {
		tmp = 2.0 * (t_1 * (cos(k) / (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * (1.0d0 / k)) ** 2.0d0
    if (k <= 0.0136d0) then
        tmp = 2.0d0 * (t_1 * ((((-0.058333333333333334d0) * ((k ** 2.0d0) / t)) + (1.0d0 / (t * (k ** 2.0d0)))) + (0.16666666666666666d0 * ((-1.0d0) / t))))
    else
        tmp = 2.0d0 * (t_1 * (cos(k) / (t * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((l * (1.0 / k)), 2.0);
	double tmp;
	if (k <= 0.0136) {
		tmp = 2.0 * (t_1 * (((-0.058333333333333334 * (Math.pow(k, 2.0) / t)) + (1.0 / (t * Math.pow(k, 2.0)))) + (0.16666666666666666 * (-1.0 / t))));
	} else {
		tmp = 2.0 * (t_1 * (Math.cos(k) / (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((l * (1.0 / k)), 2.0)
	tmp = 0
	if k <= 0.0136:
		tmp = 2.0 * (t_1 * (((-0.058333333333333334 * (math.pow(k, 2.0) / t)) + (1.0 / (t * math.pow(k, 2.0)))) + (0.16666666666666666 * (-1.0 / t))))
	else:
		tmp = 2.0 * (t_1 * (math.cos(k) / (t * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(1.0 / k)) ^ 2.0
	tmp = 0.0
	if (k <= 0.0136)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(Float64(-0.058333333333333334 * Float64((k ^ 2.0) / t)) + Float64(1.0 / Float64(t * (k ^ 2.0)))) + Float64(0.16666666666666666 * Float64(-1.0 / t)))));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(cos(k) / Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l * (1.0 / k)) ^ 2.0;
	tmp = 0.0;
	if (k <= 0.0136)
		tmp = 2.0 * (t_1 * (((-0.058333333333333334 * ((k ^ 2.0) / t)) + (1.0 / (t * (k ^ 2.0)))) + (0.16666666666666666 * (-1.0 / t))));
	else
		tmp = 2.0 * (t_1 * (cos(k) / (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 0.0136], N[(2.0 * N[(t$95$1 * N[(N[(N[(-0.058333333333333334 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0135999999999999992

   1. Initial program 44.5%

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

      1. associate-*l* 44.5%

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

      2. associate-/r* 44.5%

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

      3. sub-neg 44.5%

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

      4. distribute-rgt-in 39.4%

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

      5. unpow2 39.4%

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

      6. times-frac 27.8%

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

      7. sqr-neg 27.8%

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

      8. times-frac 39.4%

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

      9. unpow2 39.4%

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

      10. distribute-rgt-in 44.5%

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

      11. +-commutative 44.5%

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

      12. associate-+l+ 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in t around 0 78.9%

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

      1. times-frac 83.1%

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

   7. Simplified 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. pow2 83.1%

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

      3. div-inv 83.0%

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

      4. sqrt-prod 83.0%

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

      5. unpow2 83.0%

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

      6. sqrt-prod 41.2%

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

      7. add-sqr-sqrt 90.3%

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

      8. pow-flip 90.3%

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

      9. metadata-eval 90.3%

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

   9. Applied egg-rr 90.3%

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

   10. Taylor expanded in k around 0 96.3%

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

   11. Taylor expanded in k around 0 65.5%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \color{blue}{\left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

   if 0.0135999999999999992 < k

   1. Initial program 24.2%

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

      1. associate-*l* 24.1%

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

      2. associate-/r* 24.1%

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

      3. sub-neg 24.1%

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

      4. distribute-rgt-in 22.7%

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

      5. unpow2 22.7%

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

      6. times-frac 20.0%

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

      7. sqr-neg 20.0%

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

      8. times-frac 22.7%

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

      9. unpow2 22.7%

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

      10. distribute-rgt-in 24.1%

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

      11. +-commutative 24.1%

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

      12. associate-+l+ 29.4%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in t around 0 69.9%

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

      1. times-frac 67.2%

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

   7. Simplified 67.2%

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

      1. add-sqr-sqrt 67.2%

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

      2. pow2 67.2%

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

      3. div-inv 67.1%

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

      4. sqrt-prod 67.1%

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

      5. unpow2 67.1%

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

      6. sqrt-prod 43.1%

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

      7. add-sqr-sqrt 72.6%

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

      8. pow-flip 72.6%

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

      9. metadata-eval 72.6%

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

   9. Applied egg-rr 72.6%

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

   10. Taylor expanded in k around 0 93.0%

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

       1. unpow2 93.0%

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

       2. sin-mult 92.8%

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

   12. Applied egg-rr 92.8%

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

       1. div-sub 92.8%

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

       2. +-inverses 92.8%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

       3. cos-0 92.8%

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

       4. metadata-eval 92.8%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

       5. count-2 92.8%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}\right) \]

       6. *-commutative 92.8%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}\right) \]

   14. Simplified 92.8%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0136:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\left(-0.058333333333333334 \cdot \frac{{k}^{2}}{t} + \frac{1}{t \cdot {k}^{2}}\right) + 0.16666666666666666 \cdot \frac{-1}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\ell \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{if}\;k \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \frac{\frac{1}{t}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (* l (/ 1.0 k)) 2.0)))
   (if (<= k 7.5e-5)
     (* 2.0 (* t_1 (/ (/ 1.0 t) (pow k 2.0))))
     (* 2.0 (* t_1 (/ (cos k) (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((l * (1.0 / k)), 2.0);
	double tmp;
	if (k <= 7.5e-5) {
		tmp = 2.0 * (t_1 * ((1.0 / t) / pow(k, 2.0)));
	} else {
		tmp = 2.0 * (t_1 * (cos(k) / (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l * (1.0d0 / k)) ** 2.0d0
    if (k <= 7.5d-5) then
        tmp = 2.0d0 * (t_1 * ((1.0d0 / t) / (k ** 2.0d0)))
    else
        tmp = 2.0d0 * (t_1 * (cos(k) / (t * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((l * (1.0 / k)), 2.0);
	double tmp;
	if (k <= 7.5e-5) {
		tmp = 2.0 * (t_1 * ((1.0 / t) / Math.pow(k, 2.0)));
	} else {
		tmp = 2.0 * (t_1 * (Math.cos(k) / (t * (0.5 - (Math.cos((2.0 * k)) / 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((l * (1.0 / k)), 2.0)
	tmp = 0
	if k <= 7.5e-5:
		tmp = 2.0 * (t_1 * ((1.0 / t) / math.pow(k, 2.0)))
	else:
		tmp = 2.0 * (t_1 * (math.cos(k) / (t * (0.5 - (math.cos((2.0 * k)) / 2.0)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(1.0 / k)) ^ 2.0
	tmp = 0.0
	if (k <= 7.5e-5)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(1.0 / t) / (k ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(cos(k) / Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l * (1.0 / k)) ^ 2.0;
	tmp = 0.0;
	if (k <= 7.5e-5)
		tmp = 2.0 * (t_1 * ((1.0 / t) / (k ^ 2.0)));
	else
		tmp = 2.0 * (t_1 * (cos(k) / (t * (0.5 - (cos((2.0 * k)) / 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 7.5e-5], N[(2.0 * N[(t$95$1 * N[(N[(1.0 / t), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999934e-5

   1. Initial program 44.6%

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

      1. associate-*l* 44.6%

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

      2. associate-/r* 44.6%

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

      3. sub-neg 44.6%

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

      4. distribute-rgt-in 39.5%

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

      5. unpow2 39.5%

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

      6. times-frac 27.7%

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

      7. sqr-neg 27.7%

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

      8. times-frac 39.5%

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

      9. unpow2 39.5%

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

      10. distribute-rgt-in 44.6%

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

      11. +-commutative 44.6%

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

      12. associate-+l+ 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in t around 0 78.6%

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

      1. times-frac 82.8%

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

   7. Simplified 82.8%

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

      1. add-sqr-sqrt 82.8%

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

      2. pow2 82.8%

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

      3. div-inv 82.8%

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

      4. sqrt-prod 82.8%

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

      5. unpow2 82.8%

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

      6. sqrt-prod 40.8%

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

      7. add-sqr-sqrt 90.1%

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

      8. pow-flip 90.1%

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

      9. metadata-eval 90.1%

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

   9. Applied egg-rr 90.1%

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

   10. Taylor expanded in k around 0 96.3%

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

   11. Taylor expanded in k around 0 82.6%

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

       1. *-commutative 82.6%

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

       2. associate-/r* 82.6%

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

   13. Simplified 82.6%

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

   if 7.49999999999999934e-5 < k

   1. Initial program 24.5%

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

      1. associate-*l* 24.5%

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

      2. associate-/r* 24.5%

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

      3. sub-neg 24.5%

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

      4. distribute-rgt-in 23.1%

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

      5. unpow2 23.1%

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

      6. times-frac 20.5%

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

      7. sqr-neg 20.5%

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

      8. times-frac 23.1%

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

      9. unpow2 23.1%

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

      10. distribute-rgt-in 24.5%

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

      11. +-commutative 24.5%

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

      12. associate-+l+ 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in t around 0 71.0%

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

      1. times-frac 68.4%

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

   7. Simplified 68.4%

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

      1. add-sqr-sqrt 68.4%

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

      2. pow2 68.4%

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

      3. div-inv 68.4%

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

      4. sqrt-prod 68.4%

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

      5. unpow2 68.4%

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

      6. sqrt-prod 44.0%

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

      7. add-sqr-sqrt 73.6%

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

      8. pow-flip 73.6%

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

      9. metadata-eval 73.6%

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

   9. Applied egg-rr 73.6%

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

   10. Taylor expanded in k around 0 93.2%

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

       1. unpow2 93.2%

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

       2. sin-mult 92.9%

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

   12. Applied egg-rr 92.9%

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

       1. div-sub 92.9%

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

       2. +-inverses 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

       3. cos-0 92.9%

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

       4. metadata-eval 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}\right) \]

       5. count-2 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}\right) \]

       6. *-commutative 92.9%

          \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}\right) \]

   14. Simplified 92.9%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\frac{1}{t}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} + 0.16666666666666666 \cdot \frac{-1}{t}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (*
   (pow (* l (/ 1.0 k)) 2.0)
   (+ (/ 1.0 (* t (pow k 2.0))) (* 0.16666666666666666 (/ -1.0 t))))))

C

double code(double t, double l, double k) {
	return 2.0 * (pow((l * (1.0 / k)), 2.0) * ((1.0 / (t * pow(k, 2.0))) + (0.16666666666666666 * (-1.0 / t))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l * (1.0d0 / k)) ** 2.0d0) * ((1.0d0 / (t * (k ** 2.0d0))) + (0.16666666666666666d0 * ((-1.0d0) / t))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (Math.pow((l * (1.0 / k)), 2.0) * ((1.0 / (t * Math.pow(k, 2.0))) + (0.16666666666666666 * (-1.0 / t))));
}

Python

def code(t, l, k):
	return 2.0 * (math.pow((l * (1.0 / k)), 2.0) * ((1.0 / (t * math.pow(k, 2.0))) + (0.16666666666666666 * (-1.0 / t))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64((Float64(l * Float64(1.0 / k)) ^ 2.0) * Float64(Float64(1.0 / Float64(t * (k ^ 2.0))) + Float64(0.16666666666666666 * Float64(-1.0 / t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l * (1.0 / k)) ^ 2.0) * ((1.0 / (t * (k ^ 2.0))) + (0.16666666666666666 * (-1.0 / t))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[Power[N[(l * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-/r* 38.5%

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

   3. sub-neg 38.5%

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

   4. distribute-rgt-in 34.5%

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

   5. unpow2 34.5%

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

   6. times-frac 25.5%

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

   7. sqr-neg 25.5%

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

   8. times-frac 34.5%

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

   9. unpow2 34.5%

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

   10. distribute-rgt-in 38.5%

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

   11. +-commutative 38.5%

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

   12. associate-+l+ 43.5%

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

3. Simplified 43.5%

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

5. Taylor expanded in t around 0 76.3%

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

   1. times-frac 78.4%

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

7. Simplified 78.4%

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

   1. add-sqr-sqrt 78.4%

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

   2. pow2 78.4%

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

   3. div-inv 78.4%

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

   4. sqrt-prod 78.4%

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

   5. unpow2 78.4%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 85.1%

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

   8. pow-flip 85.1%

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

   9. metadata-eval 85.1%

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

9. Applied egg-rr 85.1%

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

10. Taylor expanded in k around 0 95.3%

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

11. Taylor expanded in k around 0 75.7%

    \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

12. Final simplification 75.7%

    \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\frac{1}{t \cdot {k}^{2}} + 0.16666666666666666 \cdot \frac{-1}{t}\right)\right) \]
13. Add Preprocessing

Alternative 7: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (*
   (pow (* l (/ 1.0 k)) 2.0)
   (- (/ (/ 1.0 (pow k 2.0)) t) (/ 0.16666666666666666 t)))))

C

double code(double t, double l, double k) {
	return 2.0 * (pow((l * (1.0 / k)), 2.0) * (((1.0 / pow(k, 2.0)) / t) - (0.16666666666666666 / t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l * (1.0d0 / k)) ** 2.0d0) * (((1.0d0 / (k ** 2.0d0)) / t) - (0.16666666666666666d0 / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (Math.pow((l * (1.0 / k)), 2.0) * (((1.0 / Math.pow(k, 2.0)) / t) - (0.16666666666666666 / t)));
}

Python

def code(t, l, k):
	return 2.0 * (math.pow((l * (1.0 / k)), 2.0) * (((1.0 / math.pow(k, 2.0)) / t) - (0.16666666666666666 / t)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64((Float64(l * Float64(1.0 / k)) ^ 2.0) * Float64(Float64(Float64(1.0 / (k ^ 2.0)) / t) - Float64(0.16666666666666666 / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l * (1.0 / k)) ^ 2.0) * (((1.0 / (k ^ 2.0)) / t) - (0.16666666666666666 / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[Power[N[(l * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-/r* 38.5%

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

   3. sub-neg 38.5%

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

   4. distribute-rgt-in 34.5%

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

   5. unpow2 34.5%

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

   6. times-frac 25.5%

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

   7. sqr-neg 25.5%

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

   8. times-frac 34.5%

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

   9. unpow2 34.5%

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

   10. distribute-rgt-in 38.5%

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

   11. +-commutative 38.5%

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

   12. associate-+l+ 43.5%

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

3. Simplified 43.5%

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

5. Taylor expanded in t around 0 76.3%

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

   1. times-frac 78.4%

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

7. Simplified 78.4%

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

   1. add-sqr-sqrt 78.4%

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

   2. pow2 78.4%

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

   3. div-inv 78.4%

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

   4. sqrt-prod 78.4%

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

   5. unpow2 78.4%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 85.1%

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

   8. pow-flip 85.1%

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

   9. metadata-eval 85.1%

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

9. Applied egg-rr 85.1%

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

10. Taylor expanded in k around 0 95.3%

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

11. Taylor expanded in k around 0 75.7%

    \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
12. Step-by-step derivation

    1. associate-/r* 75.7%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

    2. associate-*r/ 75.7%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

    3. metadata-eval 75.7%

       \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

13. Simplified 75.7%

    \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)}\right) \]

14. Final simplification 75.7%

    \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \left(\frac{\frac{1}{{k}^{2}}}{t} - \frac{0.16666666666666666}{t}\right)\right) \]
15. Add Preprocessing

Alternative 8: 69.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (pow (* l (/ 1.0 k)) 2.0) (/ (/ 1.0 t) (pow k 2.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. associate-*l* 38.5%

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

   2. associate-/r* 38.5%

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

   3. sub-neg 38.5%

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

   4. distribute-rgt-in 34.5%

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

   5. unpow2 34.5%

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

   6. times-frac 25.5%

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

   7. sqr-neg 25.5%

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

   8. times-frac 34.5%

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

   9. unpow2 34.5%

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

   10. distribute-rgt-in 38.5%

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

   11. +-commutative 38.5%

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

   12. associate-+l+ 43.5%

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

3. Simplified 43.5%

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

5. Taylor expanded in t around 0 76.3%

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

   1. times-frac 78.4%

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

7. Simplified 78.4%

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

   1. add-sqr-sqrt 78.4%

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

   2. pow2 78.4%

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

   3. div-inv 78.4%

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

   4. sqrt-prod 78.4%

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

   5. unpow2 78.4%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 85.1%

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

   8. pow-flip 85.1%

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

   9. metadata-eval 85.1%

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

9. Applied egg-rr 85.1%

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

10. Taylor expanded in k around 0 95.3%

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

11. Taylor expanded in k around 0 73.3%

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

    1. *-commutative 73.3%

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

    2. associate-/r* 73.3%

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

13. Simplified 73.3%

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

14. Final simplification 73.3%

    \[\leadsto 2 \cdot \left({\left(\ell \cdot \frac{1}{k}\right)}^{2} \cdot \frac{\frac{1}{t}}{{k}^{2}}\right) \]
15. Add Preprocessing

Alternative 9: 59.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (/ (pow l 2.0) t) (* 2.0 (pow k -4.0))))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l ** 2.0d0) / t) * (2.0d0 * (k ** (-4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in k around 0 65.4%

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

   1. associate-/l* 63.7%

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

5. Simplified 63.7%

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

6. Taylor expanded in k around 0 65.4%

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

   1. associate-*r/ 65.4%

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

   2. times-frac 63.7%

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

   3. *-commutative 63.7%

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

8. Simplified 63.7%

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

   1. add-log-exp 62.0%

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

   2. *-commutative 62.0%

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

   3. exp-prod 62.5%

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

   4. div-inv 62.5%

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

   5. pow-flip 62.5%

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

   6. metadata-eval 62.5%

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

   7. metadata-eval 62.5%

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

   8. pow-prod-up 62.5%

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

   9. exp-prod 62.5%

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

   10. pow-prod-up 62.5%

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

   11. metadata-eval 62.5%

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

10. Applied egg-rr 62.5%

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

    1. log-pow 60.4%

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

    2. log-pow 63.7%

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

    3. rem-log-exp 63.7%

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

    4. *-commutative 63.7%

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

12. Simplified 63.7%

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

13. Final simplification 63.7%

    \[\leadsto \frac{{\ell}^{2}}{t} \cdot \left(2 \cdot {k}^{-4}\right) \]
14. Add Preprocessing

Alternative 10: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (pow l 2.0) (pow k 4.0)) (/ 2.0 t)))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l ** 2.0d0) / (k ** 4.0d0)) * (2.0d0 / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

3. Taylor expanded in k around 0 65.4%

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

   1. associate-/l* 63.7%

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

5. Simplified 63.7%

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

6. Taylor expanded in k around 0 65.4%

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

   1. associate-*r/ 65.4%

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

   2. times-frac 63.7%

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

   3. *-commutative 63.7%

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

8. Simplified 63.7%

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

   1. add-log-exp 62.0%

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

   2. *-commutative 62.0%

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

   3. exp-prod 62.5%

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

   4. div-inv 62.5%

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

   5. pow-flip 62.5%

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

   6. metadata-eval 62.5%

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

   7. metadata-eval 62.5%

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

   8. pow-prod-up 62.5%

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

   9. exp-prod 62.5%

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

   10. pow-prod-up 62.5%

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

   11. metadata-eval 62.5%

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

10. Applied egg-rr 62.5%

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

    1. log-pow 60.4%

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

    2. log-pow 63.7%

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

    3. rem-log-exp 63.7%

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

    4. *-commutative 63.7%

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

12. Simplified 63.7%

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

13. Taylor expanded in l around 0 65.4%

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

    1. associate-*r/ 65.4%

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

    2. *-commutative 65.4%

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

    3. times-frac 66.1%

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

15. Simplified 66.1%

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

16. Final simplification 66.1%

    \[\leadsto \frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))