Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 82.4%

Time: 20.6s

Alternatives: 17

Speedup: 24.7×

• 26.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(2 + t_1\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        2e-19)
     (* (/ 2.0 (* (tan k) (+ 2.0 t_1))) (/ (* l l) (* (pow t 3.0) (sin k))))
     (/ 2.0 (* (* t (/ k l)) (* k (/ (pow (sin k) 2.0) (* l (cos k)))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e-19) {
		tmp = (2.0 / (tan(k) * (2.0 + t_1))) * ((l * l) / (pow(t, 3.0) * sin(k)));
	} else {
		tmp = 2.0 / ((t * (k / l)) * (k * (pow(sin(k), 2.0) / (l * cos(k)))));
	}
	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 = (k / t) ** 2.0d0
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + t_1)))) <= 2d-19) then
        tmp = (2.0d0 / (tan(k) * (2.0d0 + t_1))) * ((l * l) / ((t ** 3.0d0) * sin(k)))
    else
        tmp = 2.0d0 / ((t * (k / l)) * (k * ((sin(k) ** 2.0d0) / (l * cos(k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e-19) {
		tmp = (2.0 / (Math.tan(k) * (2.0 + t_1))) * ((l * l) / (Math.pow(t, 3.0) * Math.sin(k)));
	} else {
		tmp = 2.0 / ((t * (k / l)) * (k * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e-19:
		tmp = (2.0 / (math.tan(k) * (2.0 + t_1))) * ((l * l) / (math.pow(t, 3.0) * math.sin(k)))
	else:
		tmp = 2.0 / ((t * (k / l)) * (k * (math.pow(math.sin(k), 2.0) / (l * math.cos(k)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 2e-19)
		tmp = Float64(Float64(2.0 / Float64(tan(k) * Float64(2.0 + t_1))) * Float64(Float64(l * l) / Float64((t ^ 3.0) * sin(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64(k * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e-19)
		tmp = (2.0 / (tan(k) * (2.0 + t_1))) * ((l * l) / ((t ^ 3.0) * sin(k)));
	else
		tmp = 2.0 / ((t * (k / l)) * (k * ((sin(k) ^ 2.0) / (l * cos(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-19], N[(N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2e-19

   1. Initial program 79.8%

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

      1. associate-/l/ 79.8%

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

      2. associate-*l/ 80.5%

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

      3. associate-*l/ 79.6%

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

      4. associate-/r/ 79.6%

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

      5. *-commutative 79.6%

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

      6. associate-/l/ 79.7%

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

      7. associate-*r* 79.8%

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

      8. *-commutative 79.8%

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

      9. associate-*r* 79.7%

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

      10. *-commutative 79.7%

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

   3. Simplified 79.7%

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

      1. expm1-log1p-u 62.1%

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

      2. expm1-udef 55.3%

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

      3. *-commutative 55.3%

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

   5. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 62.1%

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

      2. expm1-log1p 79.7%

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

      3. unpow2 79.7%

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

      4. *-commutative 79.7%

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

      5. associate-*l/ 79.7%

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

      6. associate-*r* 79.7%

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

      7. *-commutative 79.7%

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

      8. times-frac 80.5%

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

      9. unpow2 80.5%

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

      10. *-commutative 80.5%

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

   7. Simplified 80.5%

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

   if 2e-19 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-*l* 19.4%

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

      3. associate-*r* 19.4%

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

      4. +-commutative 19.4%

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

      5. associate-+r+ 19.4%

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

      6. metadata-eval 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in k around inf 47.5%

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

      1. *-commutative 47.5%

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

      2. unpow2 47.5%

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

      3. times-frac 57.0%

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

      4. unpow2 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in t around 0 47.5%

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

      1. times-frac 46.6%

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

      2. *-commutative 46.6%

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

      3. *-commutative 46.6%

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

      4. associate-/l* 46.7%

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

      5. times-frac 47.5%

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

      6. *-commutative 47.5%

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

      7. associate-/r/ 47.5%

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

      8. unpow2 47.5%

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

      9. associate-/l* 50.9%

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

      10. associate-/r/ 59.1%

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

      11. associate-*r/ 57.8%

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

      12. unpow2 57.8%

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

      13. associate-*l* 66.9%

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

      14. associate-/l/ 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in k around 0 75.9%

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

       1. associate-*r/ 75.9%

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

       2. associate-/l* 67.0%

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

   12. Applied egg-rr 67.0%

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

       1. associate-*r/ 67.0%

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

       2. *-commutative 67.0%

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

       3. associate-*l* 73.2%

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

       4. associate-/r/ 81.9%

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

       5. *-commutative 81.9%

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

   14. Simplified 81.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}\\ \end{array} \]

Alternative 2: 71.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-33)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (/ (pow (sin k) 2.0) (* l (cos k))) (* k (* t (/ k l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-33) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * cos(k))) * (k * (t * (k / l))));
	}
	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 <= 2.7d-33) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / (l * cos(k))) * (k * (t * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-33) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))) * (k * (t * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-33:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / (l * math.cos(k))) * (k * (t * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-33)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / Float64(l * cos(k))) * Float64(k * Float64(t * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-33)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / (l * cos(k))) * (k * (t * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-33], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-33

   1. Initial program 52.0%

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

      1. associate-/l/ 52.0%

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

      2. associate-*l/ 52.0%

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

      3. associate-*l/ 50.9%

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

      4. associate-/r/ 50.9%

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

      5. *-commutative 50.9%

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

      6. associate-/l/ 50.9%

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

      7. associate-*r* 51.4%

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

      8. *-commutative 51.4%

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

      9. associate-*r* 51.4%

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

      10. *-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in k around 0 48.5%

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

   5. Taylor expanded in k around 0 45.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 45.3%

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

      2. associate-*l* 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in l around 0 45.3%

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

      1. unpow2 45.3%

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

      2. unpow2 45.3%

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

      3. associate-*r* 48.5%

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

      4. times-frac 55.1%

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

      5. associate-/r* 56.1%

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

   10. Simplified 56.1%

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

   if 2.7000000000000001e-33 < k

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

      2. associate-*l* 52.3%

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

      3. associate-*r* 52.3%

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

      4. +-commutative 52.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in k around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. unpow2 65.3%

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

      3. times-frac 65.6%

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

      4. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in t around 0 65.3%

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

      1. times-frac 62.8%

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

      2. *-commutative 62.8%

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

      3. *-commutative 62.8%

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

      4. associate-/l* 62.8%

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

      5. times-frac 65.3%

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

      6. *-commutative 65.3%

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

      7. associate-/r/ 65.2%

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

      8. unpow2 65.2%

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

      9. associate-/l* 65.3%

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

      10. associate-/r/ 70.9%

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

      11. associate-*r/ 68.0%

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

      12. unpow2 68.0%

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

      13. associate-*l* 75.8%

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

      14. associate-/l/ 75.8%

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

   9. Simplified 75.8%

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

   10. Taylor expanded in k around 0 81.1%

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

   11. Taylor expanded in k around 0 70.9%

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

       1. associate-/l* 68.0%

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

       2. unpow2 68.0%

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

       3. associate-*r/ 75.8%

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

       4. associate-/r/ 80.8%

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

   13. Simplified 80.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 3: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \left(k \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-33)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (/ (pow (sin k) 2.0) (* l (cos k))) (* k (/ (* t k) l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-33) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * cos(k))) * (k * ((t * k) / l)));
	}
	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 <= 2.5d-33) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / (l * cos(k))) * (k * ((t * k) / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-33) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k))) * (k * ((t * k) / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.5e-33:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / (l * math.cos(k))) * (k * ((t * k) / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-33)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / Float64(l * cos(k))) * Float64(k * Float64(Float64(t * k) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-33)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) / (l * cos(k))) * (k * ((t * k) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.5e-33], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000014e-33

   1. Initial program 52.0%

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

      1. associate-/l/ 52.0%

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

      2. associate-*l/ 52.0%

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

      3. associate-*l/ 50.9%

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

      4. associate-/r/ 50.9%

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

      5. *-commutative 50.9%

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

      6. associate-/l/ 50.9%

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

      7. associate-*r* 51.4%

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

      8. *-commutative 51.4%

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

      9. associate-*r* 51.4%

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

      10. *-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in k around 0 48.5%

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

   5. Taylor expanded in k around 0 45.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 45.3%

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

      2. associate-*l* 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in l around 0 45.3%

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

      1. unpow2 45.3%

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

      2. unpow2 45.3%

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

      3. associate-*r* 48.5%

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

      4. times-frac 55.1%

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

      5. associate-/r* 56.1%

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

   10. Simplified 56.1%

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

   if 2.50000000000000014e-33 < k

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

      2. associate-*l* 52.3%

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

      3. associate-*r* 52.3%

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

      4. +-commutative 52.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in k around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. unpow2 65.3%

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

      3. times-frac 65.6%

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

      4. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in t around 0 65.3%

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

      1. times-frac 62.8%

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

      2. *-commutative 62.8%

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

      3. *-commutative 62.8%

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

      4. associate-/l* 62.8%

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

      5. times-frac 65.3%

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

      6. *-commutative 65.3%

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

      7. associate-/r/ 65.2%

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

      8. unpow2 65.2%

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

      9. associate-/l* 65.3%

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

      10. associate-/r/ 70.9%

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

      11. associate-*r/ 68.0%

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

      12. unpow2 68.0%

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

      13. associate-*l* 75.8%

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

      14. associate-/l/ 75.8%

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

   9. Simplified 75.8%

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

   10. Taylor expanded in k around 0 81.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \cos k} \cdot \left(k \cdot \frac{t \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 4: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.5e-33)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (* t (/ k l)) (* k (/ (pow (sin k) 2.0) (* l (cos k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-33) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t * (k / l)) * (k * (pow(sin(k), 2.0) / (l * cos(k)))));
	}
	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 <= 2.5d-33) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / ((t * (k / l)) * (k * ((sin(k) ** 2.0d0) / (l * cos(k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.5e-33) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t * (k / l)) * (k * (Math.pow(Math.sin(k), 2.0) / (l * Math.cos(k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.5e-33:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((t * (k / l)) * (k * (math.pow(math.sin(k), 2.0) / (l * math.cos(k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.5e-33)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) * Float64(k * Float64((sin(k) ^ 2.0) / Float64(l * cos(k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.5e-33)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / ((t * (k / l)) * (k * ((sin(k) ^ 2.0) / (l * cos(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.5e-33], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000014e-33

   1. Initial program 52.0%

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

      1. associate-/l/ 52.0%

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

      2. associate-*l/ 52.0%

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

      3. associate-*l/ 50.9%

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

      4. associate-/r/ 50.9%

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

      5. *-commutative 50.9%

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

      6. associate-/l/ 50.9%

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

      7. associate-*r* 51.4%

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

      8. *-commutative 51.4%

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

      9. associate-*r* 51.4%

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

      10. *-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in k around 0 48.5%

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

   5. Taylor expanded in k around 0 45.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 45.3%

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

      2. associate-*l* 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in l around 0 45.3%

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

      1. unpow2 45.3%

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

      2. unpow2 45.3%

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

      3. associate-*r* 48.5%

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

      4. times-frac 55.1%

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

      5. associate-/r* 56.1%

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

   10. Simplified 56.1%

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

   if 2.50000000000000014e-33 < k

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

      2. associate-*l* 52.3%

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

      3. associate-*r* 52.3%

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

      4. +-commutative 52.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in k around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. unpow2 65.3%

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

      3. times-frac 65.6%

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

      4. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in t around 0 65.3%

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

      1. times-frac 62.8%

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

      2. *-commutative 62.8%

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

      3. *-commutative 62.8%

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

      4. associate-/l* 62.8%

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

      5. times-frac 65.3%

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

      6. *-commutative 65.3%

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

      7. associate-/r/ 65.2%

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

      8. unpow2 65.2%

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

      9. associate-/l* 65.3%

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

      10. associate-/r/ 70.9%

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

      11. associate-*r/ 68.0%

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

      12. unpow2 68.0%

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

      13. associate-*l* 75.8%

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

      14. associate-/l/ 75.8%

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

   9. Simplified 75.8%

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

   10. Taylor expanded in k around 0 81.1%

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

       1. associate-*r/ 81.2%

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

       2. associate-/l* 75.8%

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

   12. Applied egg-rr 75.8%

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

       1. associate-*r/ 75.8%

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

       2. *-commutative 75.8%

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

       3. associate-*l* 80.7%

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

       4. associate-/r/ 85.7%

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

       5. *-commutative 85.7%

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

   14. Simplified 85.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}\right)}\\ \end{array} \]

Alternative 5: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.042:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+131}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.042)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 9.5e+131)
     (* (* l l) (/ 2.0 (* (tan k) (* t (* (sin k) (* k k))))))
     (*
      2.0
      (*
       (* (/ l k) (/ l k))
       (+ (/ 1.0 (* t (* k k))) (+ (/ 0.3333333333333333 t) (/ -0.5 t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.042) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 9.5e+131) {
		tmp = (l * l) * (2.0 / (tan(k) * (t * (sin(k) * (k * k)))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	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 <= 0.042d0) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 9.5d+131) then
        tmp = (l * l) * (2.0d0 / (tan(k) * (t * (sin(k) * (k * k)))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((1.0d0 / (t * (k * k))) + ((0.3333333333333333d0 / t) + ((-0.5d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.042) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 9.5e+131) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * (t * (Math.sin(k) * (k * k)))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.042:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 9.5e+131:
		tmp = (l * l) * (2.0 / (math.tan(k) * (t * (math.sin(k) * (k * k)))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.042)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 9.5e+131)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(t * Float64(sin(k) * Float64(k * k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(Float64(0.3333333333333333 / t) + Float64(-0.5 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.042)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 9.5e+131)
		tmp = (l * l) * (2.0 / (tan(k) * (t * (sin(k) * (k * k)))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.042], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.5e+131], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.0420000000000000026

   1. Initial program 53.5%

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

      1. associate-/l/ 53.5%

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

      2. associate-*l/ 53.5%

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

      3. associate-*l/ 52.4%

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

      4. associate-/r/ 52.4%

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

      5. *-commutative 52.4%

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

      6. associate-/l/ 52.4%

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

      7. associate-*r* 53.0%

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

      8. *-commutative 53.0%

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

      9. associate-*r* 53.0%

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

      10. *-commutative 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 50.2%

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

   5. Taylor expanded in k around 0 47.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 47.2%

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

      2. associate-*l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in l around 0 47.2%

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

      1. unpow2 47.2%

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

      2. unpow2 47.2%

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

      3. associate-*r* 50.2%

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

      4. times-frac 56.4%

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

      5. associate-/r* 57.4%

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

   10. Simplified 57.4%

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

   if 0.0420000000000000026 < k < 9.50000000000000015e131

   1. Initial program 32.1%

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

      1. associate-/l/ 32.1%

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

      2. associate-*l/ 32.1%

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

      3. associate-*l/ 32.1%

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

      4. associate-/r/ 32.1%

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

      5. *-commutative 32.1%

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

      6. associate-/l/ 32.1%

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

      7. associate-*r* 32.1%

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

      8. *-commutative 32.1%

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

      9. associate-*r* 32.1%

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

      10. *-commutative 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in k around inf 64.4%

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

      1. associate-*r* 64.4%

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

      2. unpow2 64.4%

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

   6. Simplified 64.4%

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

   if 9.50000000000000015e131 < k

   1. Initial program 54.5%

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

      1. associate-*l* 54.5%

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

      2. associate-/l/ 54.5%

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

      3. *-commutative 54.5%

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

      4. associate-*r/ 54.5%

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

      5. associate-/l* 54.3%

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

      6. associate-/r/ 54.3%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in k around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. times-frac 68.2%

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

      3. unpow2 68.2%

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

      4. associate-/r* 68.2%

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

      5. *-commutative 68.2%

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

      6. unpow2 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in k around 0 68.1%

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

      1. cancel-sign-sub-inv 68.1%

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

      2. fma-def 68.1%

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

      3. unpow2 68.1%

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

      4. associate-/r* 68.1%

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

      5. unpow2 68.1%

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

      6. associate-*r/ 68.1%

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

      7. unpow2 68.1%

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

      8. associate-/r* 68.1%

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

      9. metadata-eval 68.1%

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

      10. unpow2 68.1%

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

   9. Simplified 68.1%

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

   10. Taylor expanded in l around 0 68.2%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 68.4%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}}}} \]

       2. associate-/r/ 68.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right)} \]

       3. unpow2 68.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       4. unpow2 68.5%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       5. times-frac 71.7%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       6. sub-neg 71.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) + \left(-0.5 \cdot \frac{1}{t}\right)\right)}\right) \]

       7. +-commutative 71.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right) \]

       8. associate-+l+ 71.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)}\right) \]

       9. *-commutative 71.7%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       10. unpow2 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       11. associate-*r/ 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       12. metadata-eval 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{\color{blue}{0.3333333333333333}}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       13. associate-*r/ 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right)\right)\right) \]

       14. metadata-eval 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\frac{\color{blue}{0.5}}{t}\right)\right)\right)\right) \]

       15. distribute-neg-frac 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \color{blue}{\frac{-0.5}{t}}\right)\right)\right) \]

       16. metadata-eval 71.7%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{\color{blue}{-0.5}}{t}\right)\right)\right) \]

   12. Simplified 71.7%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.042:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+131}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \]

Alternative 6: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.041:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 0.041)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (* (/ t l) (* k (/ k l))) (* (sin k) (tan k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.041) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (((t / l) * (k * (k / l))) * (sin(k) * tan(k)));
	}
	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 <= 0.041d0) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((t / l) * (k * (k / l))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.041) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (((t / l) * (k * (k / l))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 0.041:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (((t / l) * (k * (k / l))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 0.041)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(k * Float64(k / l))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.041)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((t / l) * (k * (k / l))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 0.041], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0410000000000000017

   1. Initial program 53.5%

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

      1. associate-/l/ 53.5%

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

      2. associate-*l/ 53.5%

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

      3. associate-*l/ 52.4%

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

      4. associate-/r/ 52.4%

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

      5. *-commutative 52.4%

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

      6. associate-/l/ 52.4%

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

      7. associate-*r* 53.0%

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

      8. *-commutative 53.0%

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

      9. associate-*r* 53.0%

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

      10. *-commutative 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 50.2%

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

   5. Taylor expanded in k around 0 47.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 47.2%

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

      2. associate-*l* 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in l around 0 47.2%

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

      1. unpow2 47.2%

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

      2. unpow2 47.2%

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

      3. associate-*r* 50.2%

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

      4. times-frac 56.4%

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

      5. associate-/r* 57.4%

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

   10. Simplified 57.4%

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

   if 0.0410000000000000017 < k

   1. Initial program 48.0%

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

      1. *-commutative 48.0%

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

      2. associate-*l* 47.9%

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

      3. associate-*r* 47.9%

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

      4. +-commutative 47.9%

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

      5. associate-+r+ 47.9%

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

      6. metadata-eval 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

      3. times-frac 67.5%

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

      4. unpow2 67.5%

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

   6. Simplified 67.5%

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

   7. Taylor expanded in k around 0 67.5%

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

      1. unpow2 67.5%

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

      2. associate-/l* 75.1%

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

      3. associate-/r/ 75.1%

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

   9. Simplified 75.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.041:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 7: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.6e-33)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (/ t l) (* (sin k) (* (tan k) (* k (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e-33) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t / l) * (sin(k) * (tan(k) * (k * (k / l)))));
	}
	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 <= 2.6d-33) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / ((t / l) * (sin(k) * (tan(k) * (k * (k / l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e-33) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((t / l) * (Math.sin(k) * (Math.tan(k) * (k * (k / l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.6e-33:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((t / l) * (math.sin(k) * (math.tan(k) * (k * (k / l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.6e-33)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.6e-33)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / ((t / l) * (sin(k) * (tan(k) * (k * (k / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.6e-33], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.59999999999999994e-33

   1. Initial program 52.0%

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

      1. associate-/l/ 52.0%

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

      2. associate-*l/ 52.0%

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

      3. associate-*l/ 50.9%

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

      4. associate-/r/ 50.9%

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

      5. *-commutative 50.9%

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

      6. associate-/l/ 50.9%

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

      7. associate-*r* 51.4%

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

      8. *-commutative 51.4%

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

      9. associate-*r* 51.4%

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

      10. *-commutative 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in k around 0 48.5%

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

   5. Taylor expanded in k around 0 45.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 45.3%

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

      2. associate-*l* 48.5%

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

   7. Simplified 48.5%

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

   8. Taylor expanded in l around 0 45.3%

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

      1. unpow2 45.3%

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

      2. unpow2 45.3%

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

      3. associate-*r* 48.5%

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

      4. times-frac 55.1%

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

      5. associate-/r* 56.1%

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

   10. Simplified 56.1%

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

   if 2.59999999999999994e-33 < k

   1. Initial program 52.3%

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

      1. *-commutative 52.3%

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

      2. associate-*l* 52.3%

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

      3. associate-*r* 52.3%

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

      4. +-commutative 52.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in k around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. unpow2 65.3%

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

      3. times-frac 65.6%

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

      4. unpow2 65.6%

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

   6. Simplified 65.6%

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

      1. pow1 65.6%

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

      2. associate-*l* 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. unpow1 66.8%

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

      2. *-commutative 66.8%

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

      3. associate-*l* 66.7%

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

      4. associate-/l* 73.4%

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

      5. associate-/r/ 73.4%

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

   10. Simplified 73.4%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{k}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 8: 67.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-52} \lor \neg \left(t \leq 1.66 \cdot 10^{-13}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.5e-52) (not (<= t 1.66e-13)))
   (* l (/ (/ l k) (* (pow t 3.0) k)))
   (*
    2.0
    (*
     (* (/ l k) (/ l k))
     (+ (/ 1.0 (* t (* k k))) (+ (/ 0.3333333333333333 t) (/ -0.5 t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.5e-52) || !(t <= 1.66e-13)) {
		tmp = l * ((l / k) / (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	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 ((t <= (-6.5d-52)) .or. (.not. (t <= 1.66d-13))) then
        tmp = l * ((l / k) / ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((1.0d0 / (t * (k * k))) + ((0.3333333333333333d0 / t) + ((-0.5d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.5e-52) || !(t <= 1.66e-13)) {
		tmp = l * ((l / k) / (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -6.5e-52) or not (t <= 1.66e-13):
		tmp = l * ((l / k) / (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.5e-52) || !(t <= 1.66e-13))
		tmp = Float64(l * Float64(Float64(l / k) / Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(Float64(0.3333333333333333 / t) + Float64(-0.5 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.5e-52) || ~((t <= 1.66e-13)))
		tmp = l * ((l / k) / ((t ^ 3.0) * k));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -6.5e-52], N[Not[LessEqual[t, 1.66e-13]], $MachinePrecision]], N[(l * N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5e-52 or 1.66e-13 < t

   1. Initial program 58.5%

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

      1. associate-*l* 58.5%

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

      2. associate-/l/ 58.5%

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

      3. *-commutative 58.5%

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

      4. associate-*r/ 59.1%

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

      5. associate-/l* 58.5%

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

      6. associate-/r/ 53.4%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in k around 0 50.8%

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

      1. unpow2 50.8%

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

      2. *-commutative 50.8%

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

      3. times-frac 53.5%

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

      4. unpow2 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in l around 0 50.8%

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

      1. unpow2 50.8%

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

      2. associate-/l* 54.4%

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

      3. associate-/l* 54.4%

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

      4. associate-/r/ 54.0%

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

      5. unpow2 54.0%

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

      6. associate-/l* 58.1%

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

      7. associate-/r/ 58.1%

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

   9. Simplified 58.1%

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

   10. Taylor expanded in l around 0 50.8%

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

       1. unpow2 50.8%

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

       2. unpow2 50.8%

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

       3. associate-*r* 55.1%

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

       4. associate-/l* 58.9%

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

       5. associate-*l/ 60.0%

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

       6. *-rgt-identity 60.0%

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

       7. associate-*r/ 60.0%

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

       8. associate-/r* 60.0%

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

       9. associate-/r/ 60.0%

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

       10. *-commutative 60.0%

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

       11. associate-*r/ 60.0%

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

       12. *-rgt-identity 60.0%

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

   12. Simplified 60.0%

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

   if -6.5e-52 < t < 1.66e-13

   1. Initial program 43.2%

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

      1. associate-*l* 43.2%

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

      2. associate-/l/ 43.3%

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

      3. *-commutative 43.3%

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

      4. associate-*r/ 42.4%

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

      5. associate-/l* 43.2%

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

      6. associate-/r/ 42.3%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. times-frac 68.0%

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

      3. unpow2 68.0%

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

      4. associate-/r* 68.1%

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

      5. *-commutative 68.1%

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

      6. unpow2 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in k around 0 37.9%

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

      1. cancel-sign-sub-inv 37.9%

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

      2. fma-def 37.9%

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

      3. unpow2 37.9%

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

      4. associate-/r* 38.0%

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

      5. unpow2 38.0%

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

      6. associate-*r/ 42.7%

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

      7. unpow2 42.7%

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

      8. associate-/r* 42.7%

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

      9. metadata-eval 42.7%

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

      10. unpow2 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in l around 0 63.2%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 62.7%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}}}} \]

       2. associate-/r/ 63.4%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right)} \]

       3. unpow2 63.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       4. unpow2 63.4%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       5. times-frac 70.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       6. sub-neg 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) + \left(-0.5 \cdot \frac{1}{t}\right)\right)}\right) \]

       7. +-commutative 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right) \]

       8. associate-+l+ 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)}\right) \]

       9. *-commutative 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       10. unpow2 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       11. associate-*r/ 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       12. metadata-eval 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{\color{blue}{0.3333333333333333}}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       13. associate-*r/ 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right)\right)\right) \]

       14. metadata-eval 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\frac{\color{blue}{0.5}}{t}\right)\right)\right)\right) \]

       15. distribute-neg-frac 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \color{blue}{\frac{-0.5}{t}}\right)\right)\right) \]

       16. metadata-eval 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{\color{blue}{-0.5}}{t}\right)\right)\right) \]

   12. Simplified 70.1%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-52} \lor \neg \left(t \leq 1.66 \cdot 10^{-13}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \]

Alternative 9: 68.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-52} \lor \neg \left(t \leq 2.4 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.46e-52) (not (<= t 2.4e-13)))
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (*
    2.0
    (*
     (* (/ l k) (/ l k))
     (+ (/ 1.0 (* t (* k k))) (+ (/ 0.3333333333333333 t) (/ -0.5 t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.46e-52) || !(t <= 2.4e-13)) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	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 ((t <= (-1.46d-52)) .or. (.not. (t <= 2.4d-13))) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((1.0d0 / (t * (k * k))) + ((0.3333333333333333d0 / t) + ((-0.5d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.46e-52) || !(t <= 2.4e-13)) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.46e-52) or not (t <= 2.4e-13):
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.46e-52) || !(t <= 2.4e-13))
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(Float64(0.3333333333333333 / t) + Float64(-0.5 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.46e-52) || ~((t <= 2.4e-13)))
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.46e-52], N[Not[LessEqual[t, 2.4e-13]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.46000000000000003e-52 or 2.3999999999999999e-13 < t

   1. Initial program 58.5%

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

      1. associate-/l/ 58.5%

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

      2. associate-*l/ 59.1%

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

      3. associate-*l/ 58.4%

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

      4. associate-/r/ 58.4%

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

      5. *-commutative 58.4%

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

      6. associate-/l/ 58.4%

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

      7. associate-*r* 58.4%

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

      8. *-commutative 58.4%

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

      9. associate-*r* 58.4%

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

      10. *-commutative 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in k around 0 53.8%

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

   5. Taylor expanded in k around 0 50.8%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 50.8%

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

      2. associate-*l* 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in l around 0 50.8%

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

      1. unpow2 50.8%

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

      2. unpow2 50.8%

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

      3. associate-*r* 55.1%

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

      4. times-frac 60.6%

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

      5. associate-/r* 61.3%

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

   10. Simplified 61.3%

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

   if -1.46000000000000003e-52 < t < 2.3999999999999999e-13

   1. Initial program 43.2%

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

      1. associate-*l* 43.2%

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

      2. associate-/l/ 43.3%

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

      3. *-commutative 43.3%

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

      4. associate-*r/ 42.4%

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

      5. associate-/l* 43.2%

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

      6. associate-/r/ 42.3%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. associate-*r/ 68.2%

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

      2. times-frac 68.0%

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

      3. unpow2 68.0%

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

      4. associate-/r* 68.1%

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

      5. *-commutative 68.1%

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

      6. unpow2 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in k around 0 37.9%

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

      1. cancel-sign-sub-inv 37.9%

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

      2. fma-def 37.9%

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

      3. unpow2 37.9%

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

      4. associate-/r* 38.0%

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

      5. unpow2 38.0%

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

      6. associate-*r/ 42.7%

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

      7. unpow2 42.7%

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

      8. associate-/r* 42.7%

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

      9. metadata-eval 42.7%

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

      10. unpow2 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in l around 0 63.2%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 62.7%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}}}} \]

       2. associate-/r/ 63.4%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right)} \]

       3. unpow2 63.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       4. unpow2 63.4%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       5. times-frac 70.1%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       6. sub-neg 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) + \left(-0.5 \cdot \frac{1}{t}\right)\right)}\right) \]

       7. +-commutative 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right) \]

       8. associate-+l+ 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)}\right) \]

       9. *-commutative 70.1%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       10. unpow2 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       11. associate-*r/ 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       12. metadata-eval 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{\color{blue}{0.3333333333333333}}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       13. associate-*r/ 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right)\right)\right) \]

       14. metadata-eval 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\frac{\color{blue}{0.5}}{t}\right)\right)\right)\right) \]

       15. distribute-neg-frac 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \color{blue}{\frac{-0.5}{t}}\right)\right)\right) \]

       16. metadata-eval 70.1%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{\color{blue}{-0.5}}{t}\right)\right)\right) \]

   12. Simplified 70.1%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-52} \lor \neg \left(t \leq 2.4 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \]

Alternative 10: 61.1% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4e+15)
   (/ 2.0 (* (* k (/ k l)) (* k (* k (/ t l)))))
   (*
    2.0
    (*
     (* (/ l k) (/ l k))
     (+ (/ 1.0 (* t (* k k))) (+ (/ 0.3333333333333333 t) (/ -0.5 t)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e+15) {
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	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 <= 4d+15) then
        tmp = 2.0d0 / ((k * (k / l)) * (k * (k * (t / l))))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * ((1.0d0 / (t * (k * k))) + ((0.3333333333333333d0 / t) + ((-0.5d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e+15) {
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4e+15:
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4e+15)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k / l)) * Float64(k * Float64(k * Float64(t / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) + Float64(Float64(0.3333333333333333 / t) + Float64(-0.5 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4e+15)
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * ((1.0 / (t * (k * k))) + ((0.3333333333333333 / t) + (-0.5 / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4e+15], N[(2.0 / N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / t), $MachinePrecision] + N[(-0.5 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4e15

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*l* 49.6%

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

      3. associate-*r* 49.6%

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

      4. +-commutative 49.6%

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

      5. associate-+r+ 49.6%

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

      6. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in k around inf 51.0%

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

      1. *-commutative 51.0%

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

      2. unpow2 51.0%

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

      3. times-frac 56.9%

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

      4. unpow2 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in t around 0 49.0%

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

      1. times-frac 51.0%

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

      2. *-commutative 51.0%

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

      3. *-commutative 51.0%

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

      4. associate-/l* 51.0%

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

      5. times-frac 51.0%

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

      6. *-commutative 51.0%

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

      7. associate-/r/ 51.0%

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

      8. unpow2 51.0%

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

      9. associate-/l* 52.3%

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

      10. associate-/r/ 55.1%

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

      11. associate-*r/ 57.4%

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

      12. unpow2 57.4%

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

      13. associate-*l* 60.5%

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

      14. associate-/l/ 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in k around 0 53.5%

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

       1. unpow2 53.5%

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

       2. associate-*r/ 55.6%

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

   12. Simplified 55.6%

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

   if 4e15 < k

   1. Initial program 48.0%

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

      1. associate-*l* 48.0%

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

      2. associate-/l/ 48.0%

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

      3. *-commutative 48.0%

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

      4. associate-*r/ 48.0%

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

      5. associate-/l* 47.9%

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

      6. associate-/r/ 47.9%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in k around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. times-frac 64.2%

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

      3. unpow2 64.2%

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

      4. associate-/r* 64.3%

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

      5. *-commutative 64.3%

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

      6. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in k around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. fma-def 54.0%

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

      3. unpow2 54.0%

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

      4. associate-/r* 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*r/ 54.0%

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

      7. unpow2 54.0%

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

      8. associate-/r* 54.0%

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

      9. metadata-eval 54.0%

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

      10. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in l around 0 58.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)}{{k}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 60.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}}}} \]

       2. associate-/r/ 59.1%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right)} \]

       3. unpow2 59.1%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       4. unpow2 59.1%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       5. times-frac 61.4%

          \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) - 0.5 \cdot \frac{1}{t}\right)\right) \]

       6. sub-neg 61.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot \frac{1}{t} + \frac{1}{{k}^{2} \cdot t}\right) + \left(-0.5 \cdot \frac{1}{t}\right)\right)}\right) \]

       7. +-commutative 61.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{1}{t}\right)} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right) \]

       8. associate-+l+ 61.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)}\right) \]

       9. *-commutative 61.4%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       10. unpow2 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} + \left(0.3333333333333333 \cdot \frac{1}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       11. associate-*r/ 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{t}} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       12. metadata-eval 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{\color{blue}{0.3333333333333333}}{t} + \left(-0.5 \cdot \frac{1}{t}\right)\right)\right)\right) \]

       13. associate-*r/ 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\color{blue}{\frac{0.5 \cdot 1}{t}}\right)\right)\right)\right) \]

       14. metadata-eval 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \left(-\frac{\color{blue}{0.5}}{t}\right)\right)\right)\right) \]

       15. distribute-neg-frac 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \color{blue}{\frac{-0.5}{t}}\right)\right)\right) \]

       16. metadata-eval 61.4%

           \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{\color{blue}{-0.5}}{t}\right)\right)\right) \]

   12. Simplified 61.4%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \left(\frac{0.3333333333333333}{t} + \frac{-0.5}{t}\right)\right)\right)\\ \end{array} \]

Alternative 11: 58.5% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4e+15)
   (* (/ 2.0 (* k k)) (* (/ l t) (/ (/ l k) k)))
   (* 2.0 (* (/ (* l (/ l t)) k) (/ -0.16666666666666666 k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e+15) {
		tmp = (2.0 / (k * k)) * ((l / t) * ((l / k) / k));
	} else {
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	}
	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 <= 4d+15) then
        tmp = (2.0d0 / (k * k)) * ((l / t) * ((l / k) / k))
    else
        tmp = 2.0d0 * (((l * (l / t)) / k) * ((-0.16666666666666666d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e+15) {
		tmp = (2.0 / (k * k)) * ((l / t) * ((l / k) / k));
	} else {
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4e+15:
		tmp = (2.0 / (k * k)) * ((l / t) * ((l / k) / k))
	else:
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4e+15)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l / t) * Float64(Float64(l / k) / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / t)) / k) * Float64(-0.16666666666666666 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4e+15)
		tmp = (2.0 / (k * k)) * ((l / t) * ((l / k) / k));
	else
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4e+15], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(-0.16666666666666666 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4e15

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. associate-/l/ 53.5%

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

      3. *-commutative 53.5%

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

      4. associate-*r/ 53.6%

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

      5. associate-/l* 53.5%

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

      6. associate-/r/ 49.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in k around inf 49.0%

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

      1. associate-*r/ 49.0%

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

      2. times-frac 51.0%

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

      3. unpow2 51.0%

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

      4. associate-/r* 51.0%

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

      5. *-commutative 51.0%

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

      6. unpow2 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in k around 0 48.1%

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

      1. unpow2 48.1%

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

      2. times-frac 53.5%

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

      3. unpow2 53.5%

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

      4. associate-/r* 53.5%

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

   9. Simplified 53.5%

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

   if 4e15 < k

   1. Initial program 48.0%

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

      1. associate-*l* 48.0%

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

      2. associate-/l/ 48.0%

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

      3. *-commutative 48.0%

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

      4. associate-*r/ 48.0%

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

      5. associate-/l* 47.9%

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

      6. associate-/r/ 47.9%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in k around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. times-frac 64.2%

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

      3. unpow2 64.2%

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

      4. associate-/r* 64.3%

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

      5. *-commutative 64.3%

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

      6. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in k around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. fma-def 54.0%

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

      3. unpow2 54.0%

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

      4. associate-/r* 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*r/ 54.0%

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

      7. unpow2 54.0%

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

      8. associate-/r* 54.0%

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

      9. metadata-eval 54.0%

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

      10. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in k around inf 54.0%

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

       1. distribute-rgt-out 58.9%

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

       2. unpow2 58.9%

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

       3. associate-/l* 59.0%

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

       4. metadata-eval 59.0%

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

       5. unpow2 59.0%

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

   12. Simplified 59.0%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{k \cdot k}} \]
   13. Step-by-step derivation

       1. times-frac 60.9%

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

       2. associate-/r/ 60.9%

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

   14. Applied egg-rr 60.9%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 12: 58.9% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4e+15)
   (* (/ 2.0 (* k k)) (/ (* (/ l k) (/ l t)) k))
   (* 2.0 (* (/ (* l (/ l t)) k) (/ -0.16666666666666666 k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e+15) {
		tmp = (2.0 / (k * k)) * (((l / k) * (l / t)) / k);
	} else {
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	}
	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 <= 4d+15) then
        tmp = (2.0d0 / (k * k)) * (((l / k) * (l / t)) / k)
    else
        tmp = 2.0d0 * (((l * (l / t)) / k) * ((-0.16666666666666666d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e+15) {
		tmp = (2.0 / (k * k)) * (((l / k) * (l / t)) / k);
	} else {
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4e+15:
		tmp = (2.0 / (k * k)) * (((l / k) * (l / t)) / k)
	else:
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4e+15)
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(Float64(l / k) * Float64(l / t)) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / t)) / k) * Float64(-0.16666666666666666 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4e+15)
		tmp = (2.0 / (k * k)) * (((l / k) * (l / t)) / k);
	else
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4e+15], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(-0.16666666666666666 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4e15

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. associate-/l/ 53.5%

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

      3. *-commutative 53.5%

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

      4. associate-*r/ 53.6%

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

      5. associate-/l* 53.5%

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

      6. associate-/r/ 49.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in k around inf 49.0%

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

      1. associate-*r/ 49.0%

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

      2. times-frac 51.0%

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

      3. unpow2 51.0%

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

      4. associate-/r* 51.0%

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

      5. *-commutative 51.0%

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

      6. unpow2 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in k around 0 48.1%

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

      1. unpow2 48.1%

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

      2. times-frac 53.5%

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

      3. unpow2 53.5%

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

      4. associate-/r* 53.5%

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

   9. Simplified 53.5%

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

       1. associate-*l/ 54.0%

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

   11. Applied egg-rr 54.0%

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

   if 4e15 < k

   1. Initial program 48.0%

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

      1. associate-*l* 48.0%

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

      2. associate-/l/ 48.0%

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

      3. *-commutative 48.0%

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

      4. associate-*r/ 48.0%

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

      5. associate-/l* 47.9%

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

      6. associate-/r/ 47.9%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in k around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. times-frac 64.2%

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

      3. unpow2 64.2%

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

      4. associate-/r* 64.3%

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

      5. *-commutative 64.3%

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

      6. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in k around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. fma-def 54.0%

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

      3. unpow2 54.0%

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

      4. associate-/r* 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*r/ 54.0%

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

      7. unpow2 54.0%

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

      8. associate-/r* 54.0%

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

      9. metadata-eval 54.0%

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

      10. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in k around inf 54.0%

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

       1. distribute-rgt-out 58.9%

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

       2. unpow2 58.9%

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

       3. associate-/l* 59.0%

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

       4. metadata-eval 59.0%

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

       5. unpow2 59.0%

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

   12. Simplified 59.0%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{k \cdot k}} \]
   13. Step-by-step derivation

       1. times-frac 60.9%

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

       2. associate-/r/ 60.9%

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

   14. Applied egg-rr 60.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 13: 60.9% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.9e+15)
   (/ 2.0 (* (* k (/ k l)) (* k (* k (/ t l)))))
   (* 2.0 (* (/ (* l (/ l t)) k) (/ -0.16666666666666666 k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.9e+15) {
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))));
	} else {
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	}
	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 <= 4.9d+15) then
        tmp = 2.0d0 / ((k * (k / l)) * (k * (k * (t / l))))
    else
        tmp = 2.0d0 * (((l * (l / t)) / k) * ((-0.16666666666666666d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.9e+15) {
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))));
	} else {
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.9e+15:
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))))
	else:
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.9e+15)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(k / l)) * Float64(k * Float64(k * Float64(t / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(l / t)) / k) * Float64(-0.16666666666666666 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.9e+15)
		tmp = 2.0 / ((k * (k / l)) * (k * (k * (t / l))));
	else
		tmp = 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.9e+15], N[(2.0 / N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(-0.16666666666666666 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9e15

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*l* 49.6%

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

      3. associate-*r* 49.6%

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

      4. +-commutative 49.6%

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

      5. associate-+r+ 49.6%

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

      6. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in k around inf 51.0%

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

      1. *-commutative 51.0%

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

      2. unpow2 51.0%

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

      3. times-frac 56.9%

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

      4. unpow2 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in t around 0 49.0%

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

      1. times-frac 51.0%

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

      2. *-commutative 51.0%

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

      3. *-commutative 51.0%

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

      4. associate-/l* 51.0%

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

      5. times-frac 51.0%

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

      6. *-commutative 51.0%

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

      7. associate-/r/ 51.0%

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

      8. unpow2 51.0%

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

      9. associate-/l* 52.3%

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

      10. associate-/r/ 55.1%

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

      11. associate-*r/ 57.4%

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

      12. unpow2 57.4%

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

      13. associate-*l* 60.5%

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

      14. associate-/l/ 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in k around 0 53.5%

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

       1. unpow2 53.5%

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

       2. associate-*r/ 55.6%

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

   12. Simplified 55.6%

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

   if 4.9e15 < k

   1. Initial program 48.0%

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

      1. associate-*l* 48.0%

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

      2. associate-/l/ 48.0%

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

      3. *-commutative 48.0%

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

      4. associate-*r/ 48.0%

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

      5. associate-/l* 47.9%

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

      6. associate-/r/ 47.9%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in k around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. times-frac 64.2%

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

      3. unpow2 64.2%

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

      4. associate-/r* 64.3%

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

      5. *-commutative 64.3%

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

      6. unpow2 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in k around 0 54.0%

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

      1. cancel-sign-sub-inv 54.0%

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

      2. fma-def 54.0%

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

      3. unpow2 54.0%

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

      4. associate-/r* 54.0%

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

      5. unpow2 54.0%

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

      6. associate-*r/ 54.0%

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

      7. unpow2 54.0%

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

      8. associate-/r* 54.0%

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

      9. metadata-eval 54.0%

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

      10. unpow2 54.0%

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

   9. Simplified 54.0%

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

   10. Taylor expanded in k around inf 54.0%

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

       1. distribute-rgt-out 58.9%

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

       2. unpow2 58.9%

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

       3. associate-/l* 59.0%

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

       4. metadata-eval 59.0%

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

       5. unpow2 59.0%

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

   12. Simplified 59.0%

       \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{k \cdot k}} \]
   13. Step-by-step derivation

       1. times-frac 60.9%

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

       2. associate-/r/ 60.9%

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

   14. Applied egg-rr 60.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 14: 36.1% accurate, 32.4× speedup

Math

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

FPCore

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

C

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

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 * (l / t)) / k) * ((-0.16666666666666666d0) / k))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 * (((l * (l / t)) / k) * (-0.16666666666666666 / k))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. associate-*l* 52.1%

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

   2. associate-/l/ 52.1%

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

   3. *-commutative 52.1%

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

   4. associate-*r/ 52.1%

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

   5. associate-/l* 52.1%

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

   6. associate-/r/ 48.8%

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

3. Simplified 53.9%

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

4. Taylor expanded in k around inf 53.6%

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

   1. associate-*r/ 53.6%

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

   2. times-frac 54.3%

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

   3. unpow2 54.3%

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

   4. associate-/r* 54.4%

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

   5. *-commutative 54.4%

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

   6. unpow2 54.4%

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

6. Simplified 54.4%

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

7. Taylor expanded in k around 0 36.5%

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

   1. cancel-sign-sub-inv 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right)} \]

   2. fma-def 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   3. unpow2 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   4. associate-/r* 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   5. unpow2 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*r/ 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   7. unpow2 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   8. associate-/r* 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   9. metadata-eval 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{t}\right) \]

   10. unpow2 38.2%

       \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

9. Simplified 38.2%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)} \]

10. Taylor expanded in k around inf 31.2%

    \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 34.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 + 0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 34.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 34.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

12. Simplified 34.2%

    \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{k \cdot k}} \]
13. Step-by-step derivation

    1. times-frac 37.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

    2. associate-/r/ 37.3%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \ell}}{k} \cdot \frac{-0.16666666666666666}{k}\right) \]

14. Applied egg-rr 37.3%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{-0.16666666666666666}{k}\right)} \]

15. Final simplification 37.3%

    \[\leadsto 2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right) \]

Alternative 15: 31.0% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (/ (/ (* l l) (* k k)) t)))

C

double code(double t, double l, double k) {
	return -0.3333333333333333 * (((l * l) / (k * k)) / 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 = (-0.3333333333333333d0) * (((l * l) / (k * k)) / t)
end function

Java

public static double code(double t, double l, double k) {
	return -0.3333333333333333 * (((l * l) / (k * k)) / t);
}

Python

def code(t, l, k):
	return -0.3333333333333333 * (((l * l) / (k * k)) / t)

Julia

function code(t, l, k)
	return Float64(-0.3333333333333333 * Float64(Float64(Float64(l * l) / Float64(k * k)) / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.3333333333333333 * (((l * l) / (k * k)) / t);
end

Wolfram

code[t_, l_, k_] := N[(-0.3333333333333333 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 52.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. associate-/l/ 52.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. *-commutative 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

   4. associate-*r/ 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

   5. associate-/l* 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

   6. associate-/r/ 48.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

4. Taylor expanded in k around inf 53.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 53.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   2. times-frac 54.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}} \]

   3. unpow2 54.3%

      \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t} \]

   4. associate-/r* 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}}{t}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}}}{t} \]

   6. unpow2 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{\sin k}^{2}}}{t} \]

6. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}}{t}} \]

7. Taylor expanded in k around 0 36.5%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right)} \]

   2. fma-def 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   3. unpow2 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   4. associate-/r* 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   5. unpow2 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*r/ 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   7. unpow2 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   8. associate-/r* 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   9. metadata-eval 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{t}\right) \]

   10. unpow2 38.2%

       \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

9. Simplified 38.2%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)} \]

10. Taylor expanded in k around inf 31.2%

    \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 34.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 + 0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 34.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 34.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

12. Simplified 34.2%

    \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{k \cdot k}} \]

13. Taylor expanded in l around 0 30.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. associate-/r* 30.0%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

    2. unpow2 30.0%

       \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

    3. unpow2 30.0%

       \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

15. Simplified 30.0%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}} \]

16. Final simplification 30.0%

    \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \]

Alternative 16: 34.0% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{t \cdot k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* l l) k) (/ -0.3333333333333333 (* t k))))

C

double code(double t, double l, double k) {
	return ((l * l) / k) * (-0.3333333333333333 / (t * k));
}

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 * l) / k) * ((-0.3333333333333333d0) / (t * k))
end function

Java

public static double code(double t, double l, double k) {
	return ((l * l) / k) * (-0.3333333333333333 / (t * k));
}

Python

def code(t, l, k):
	return ((l * l) / k) * (-0.3333333333333333 / (t * k))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * l) / k) * Float64(-0.3333333333333333 / Float64(t * k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * l) / k) * (-0.3333333333333333 / (t * k));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision] * N[(-0.3333333333333333 / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 52.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. associate-/l/ 52.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. *-commutative 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

   4. associate-*r/ 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

   5. associate-/l* 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

   6. associate-/r/ 48.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

4. Taylor expanded in k around inf 53.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 53.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   2. times-frac 54.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}} \]

   3. unpow2 54.3%

      \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t} \]

   4. associate-/r* 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}}{t}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}}}{t} \]

   6. unpow2 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{\sin k}^{2}}}{t} \]

6. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}}{t}} \]

7. Taylor expanded in k around 0 36.5%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right)} \]

   2. fma-def 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   3. unpow2 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   4. associate-/r* 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   5. unpow2 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*r/ 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   7. unpow2 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   8. associate-/r* 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   9. metadata-eval 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{t}\right) \]

   10. unpow2 38.2%

       \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

9. Simplified 38.2%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)} \]

10. Taylor expanded in k around inf 31.2%

    \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
11. Step-by-step derivation

    1. distribute-rgt-out 32.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)}}{{k}^{2}} \]

    2. unpow2 32.9%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)}{{k}^{2}} \]

    3. associate-/l* 34.2%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot \left(-0.5 + 0.3333333333333333\right)}{{k}^{2}} \]

    4. metadata-eval 34.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}} \]

    5. unpow2 34.2%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{\color{blue}{k \cdot k}} \]

12. Simplified 34.2%

    \[\leadsto \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}} \cdot -0.16666666666666666}{k \cdot k}} \]

13. Taylor expanded in l around 0 30.4%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. associate-*r/ 30.4%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. *-commutative 30.4%

       \[\leadsto \frac{\color{blue}{{\ell}^{2} \cdot -0.3333333333333333}}{{k}^{2} \cdot t} \]

    3. unpow2 30.4%

       \[\leadsto \frac{{\ell}^{2} \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    4. associate-*r* 32.2%

       \[\leadsto \frac{{\ell}^{2} \cdot -0.3333333333333333}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

    5. times-frac 32.4%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k} \cdot \frac{-0.3333333333333333}{k \cdot t}} \]

    6. unpow2 32.4%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{k} \cdot \frac{-0.3333333333333333}{k \cdot t} \]

15. Simplified 32.4%

    \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{k \cdot t}} \]

16. Final simplification 32.4%

    \[\leadsto \frac{\ell \cdot \ell}{k} \cdot \frac{-0.3333333333333333}{t \cdot k} \]

Alternative 17: 32.8% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (* -0.3333333333333333 (* l (/ l t))) (* k k)))

C

double code(double t, double l, double k) {
	return (-0.3333333333333333 * (l * (l / t))) / (k * k);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((-0.3333333333333333d0) * (l * (l / t))) / (k * k)
end function

Java

public static double code(double t, double l, double k) {
	return (-0.3333333333333333 * (l * (l / t))) / (k * k);
}

Python

def code(t, l, k):
	return (-0.3333333333333333 * (l * (l / t))) / (k * k)

Julia

function code(t, l, k)
	return Float64(Float64(-0.3333333333333333 * Float64(l * Float64(l / t))) / Float64(k * k))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (-0.3333333333333333 * (l * (l / t))) / (k * k);
end

Wolfram

code[t_, l_, k_] := N[(N[(-0.3333333333333333 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 52.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. associate-/l/ 52.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

   3. *-commutative 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

   4. associate-*r/ 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

   5. associate-/l* 52.1%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

   6. associate-/r/ 48.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

3. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

4. Taylor expanded in k around inf 53.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 53.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   2. times-frac 54.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t}} \]

   3. unpow2 54.3%

      \[\leadsto \frac{2}{\color{blue}{k \cdot k}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2} \cdot t} \]

   4. associate-/r* 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}}{t}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{\sin k}^{2}}}{t} \]

   6. unpow2 54.4%

      \[\leadsto \frac{2}{k \cdot k} \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{\sin k}^{2}}}{t} \]

6. Simplified 54.4%

   \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}}{t}} \]

7. Taylor expanded in k around 0 36.5%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right)} \]

   2. fma-def 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   3. unpow2 36.5%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{\ell \cdot \ell}}{t}, \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   4. associate-/r* 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   5. unpow2 36.6%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   6. associate-*r/ 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\color{blue}{\ell \cdot \frac{\ell}{{k}^{2}}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   7. unpow2 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   8. associate-/r* 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \color{blue}{\frac{\frac{\ell}{k}}{k}}}{t}\right) + \left(--0.3333333333333333\right) \cdot \frac{{\ell}^{2}}{t}\right) \]

   9. metadata-eval 38.2%

      \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + \color{blue}{0.3333333333333333} \cdot \frac{{\ell}^{2}}{t}\right) \]

   10. unpow2 38.2%

       \[\leadsto \frac{2}{k \cdot k} \cdot \left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

9. Simplified 38.2%

   \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{\ell \cdot \ell}{t}, \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{t}\right) + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)} \]

10. Taylor expanded in k around inf 30.8%

    \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{-0.5 \cdot \frac{{\ell}^{2}}{t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right) \]
11. Step-by-step derivation

    1. unpow2 30.8%

       \[\leadsto \frac{2}{k \cdot k} \cdot \left(-0.5 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right) \]

    2. associate-/l* 32.1%

       \[\leadsto \frac{2}{k \cdot k} \cdot \left(-0.5 \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right) \]

12. Simplified 32.1%

    \[\leadsto \frac{2}{k \cdot k} \cdot \left(\color{blue}{-0.5 \cdot \frac{\ell}{\frac{t}{\ell}}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right) \]

13. Taylor expanded in k around 0 31.2%

    \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}} \]
14. Step-by-step derivation

    1. associate-*r/ 31.2%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(-0.5 \cdot \frac{{\ell}^{2}}{t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}}} \]

    2. distribute-rgt-out 32.9%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)\right)}}{{k}^{2}} \]

    3. unpow2 32.9%

       \[\leadsto \frac{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t} \cdot \left(-0.5 + 0.3333333333333333\right)\right)}{{k}^{2}} \]

    4. associate-*l/ 34.2%

       \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\frac{\ell}{t} \cdot \ell\right)} \cdot \left(-0.5 + 0.3333333333333333\right)\right)}{{k}^{2}} \]

    5. metadata-eval 34.2%

       \[\leadsto \frac{2 \cdot \left(\left(\frac{\ell}{t} \cdot \ell\right) \cdot \color{blue}{-0.16666666666666666}\right)}{{k}^{2}} \]

    6. *-commutative 34.2%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(\frac{\ell}{t} \cdot \ell\right)\right)}}{{k}^{2}} \]

    7. associate-*r* 34.2%

       \[\leadsto \frac{\color{blue}{\left(2 \cdot -0.16666666666666666\right) \cdot \left(\frac{\ell}{t} \cdot \ell\right)}}{{k}^{2}} \]

    8. metadata-eval 34.2%

       \[\leadsto \frac{\color{blue}{-0.3333333333333333} \cdot \left(\frac{\ell}{t} \cdot \ell\right)}{{k}^{2}} \]

    9. *-commutative 34.2%

       \[\leadsto \frac{-0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}}{{k}^{2}} \]

    10. unpow2 34.2%

        \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{\color{blue}{k \cdot k}} \]

15. Simplified 34.2%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}} \]

16. Final simplification 34.2%

    \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k} \]

Reproduce

herbie shell --seed 2023215 
(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))))