Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.0% → 97.1%

Time: 30.8s

Alternatives: 10

Speedup: 2.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.0% 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: 97.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k_m \cdot \frac{1}{\ell}\\ t_2 := \ell \cdot {k_m}^{-2}\\ \mathbf{if}\;k_m \leq 0.000105:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{t_2}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\cos k_m} \cdot \frac{t_1}{\frac{1}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k_m\right)}{2}\right)}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (/ 1.0 l))) (t_2 (* l (pow k_m -2.0))))
   (if (<= k_m 0.000105)
     (* 2.0 (* t_2 (/ t_2 t)))
     (/
      2.0
      (*
       (/ t_1 (cos k_m))
       (/ t_1 (/ 1.0 (* t (- 0.5 (/ (cos (* 2.0 k_m)) 2.0))))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (1.0 / l);
	double t_2 = l * pow(k_m, -2.0);
	double tmp;
	if (k_m <= 0.000105) {
		tmp = 2.0 * (t_2 * (t_2 / t));
	} else {
		tmp = 2.0 / ((t_1 / cos(k_m)) * (t_1 / (1.0 / (t * (0.5 - (cos((2.0 * k_m)) / 2.0))))));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = k_m * (1.0d0 / l)
    t_2 = l * (k_m ** (-2.0d0))
    if (k_m <= 0.000105d0) then
        tmp = 2.0d0 * (t_2 * (t_2 / t))
    else
        tmp = 2.0d0 / ((t_1 / cos(k_m)) * (t_1 / (1.0d0 / (t * (0.5d0 - (cos((2.0d0 * k_m)) / 2.0d0))))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = k_m * (1.0 / l);
	double t_2 = l * Math.pow(k_m, -2.0);
	double tmp;
	if (k_m <= 0.000105) {
		tmp = 2.0 * (t_2 * (t_2 / t));
	} else {
		tmp = 2.0 / ((t_1 / Math.cos(k_m)) * (t_1 / (1.0 / (t * (0.5 - (Math.cos((2.0 * k_m)) / 2.0))))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = k_m * (1.0 / l)
	t_2 = l * math.pow(k_m, -2.0)
	tmp = 0
	if k_m <= 0.000105:
		tmp = 2.0 * (t_2 * (t_2 / t))
	else:
		tmp = 2.0 / ((t_1 / math.cos(k_m)) * (t_1 / (1.0 / (t * (0.5 - (math.cos((2.0 * k_m)) / 2.0))))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(1.0 / l))
	t_2 = Float64(l * (k_m ^ -2.0))
	tmp = 0.0
	if (k_m <= 0.000105)
		tmp = Float64(2.0 * Float64(t_2 * Float64(t_2 / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 / cos(k_m)) * Float64(t_1 / Float64(1.0 / Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k_m)) / 2.0)))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = k_m * (1.0 / l);
	t_2 = l * (k_m ^ -2.0);
	tmp = 0.0;
	if (k_m <= 0.000105)
		tmp = 2.0 * (t_2 * (t_2 / t));
	else
		tmp = 2.0 / ((t_1 / cos(k_m)) * (t_1 / (1.0 / (t * (0.5 - (cos((2.0 * k_m)) / 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 0.000105], N[(2.0 * N[(t$95$2 * N[(t$95$2 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(1.0 / N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e-4

   1. Initial program 34.9%

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

      1. associate-*l* 35.0%

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

      2. associate-/r* 35.0%

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

      3. sub-neg 35.0%

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

      4. distribute-rgt-in 30.6%

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

      5. unpow2 30.6%

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

      6. times-frac 22.5%

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

      7. sqr-neg 22.5%

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

      8. times-frac 30.6%

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

      9. unpow2 30.6%

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

      10. distribute-rgt-in 35.0%

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

      11. +-commutative 35.0%

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

      12. associate-+l+ 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in k around 0 58.0%

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

      1. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

      1. expm1-log1p-u 57.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 55.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 55.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 57.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 57.6%

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

   11. Simplified 57.6%

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

       1. *-un-lft-identity 57.6%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{\color{blue}{1 \cdot t}} \]

       2. rem-cbrt-cube 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt[3]{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}}}}{1 \cdot t} \]

       3. unpow1/3 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{0.3333333333333333}}}{1 \cdot t} \]

       4. sqr-pow 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{1 \cdot t} \]

       5. times-frac 55.9%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right)} \]

       6. pow-pow 55.9%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \frac{0.3333333333333333}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       7. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \color{blue}{0.16666666666666666}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       8. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\color{blue}{0.5}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       9. pow1/2 55.9%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       10. sqrt-prod 55.9%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       11. unpow2 55.9%

           \[\leadsto 2 \cdot \left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       12. sqrt-prod 26.5%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       13. add-sqr-sqrt 37.7%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       14. sqrt-pow1 37.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       15. metadata-eval 37.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{\color{blue}{-2}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

   13. Applied egg-rr 74.4%

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

   if 1.05e-4 < k

   1. Initial program 31.2%

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

      1. associate-*l* 31.2%

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

      2. associate--l+ 31.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. times-frac 77.8%

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

   7. Simplified 77.8%

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

      1. associate-*r/ 77.7%

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

      2. div-inv 77.7%

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

      3. pow-flip 77.7%

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

      4. metadata-eval 77.7%

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

   9. Applied egg-rr 77.7%

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

       1. associate-/l* 77.7%

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

   11. Simplified 77.7%

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

       1. add-sqr-sqrt 77.7%

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

       2. div-inv 77.8%

          \[\leadsto \frac{2}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}} \cdot \sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\color{blue}{\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}}}} \]

       3. times-frac 77.7%

          \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}}} \]

       4. sqrt-prod 77.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       5. unpow2 77.7%

          \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       6. sqrt-prod 77.8%

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

       7. add-sqr-sqrt 77.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       8. sqrt-pow1 66.6%

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

       9. metadata-eval 66.6%

          \[\leadsto \frac{2}{\frac{k \cdot {\ell}^{\color{blue}{-1}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       10. unpow-1 66.6%

           \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\frac{1}{\ell}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       11. sqrt-prod 67.8%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       12. unpow2 67.8%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       13. sqrt-prod 69.3%

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

       14. add-sqr-sqrt 69.4%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       15. sqrt-pow1 99.0%

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

       16. metadata-eval 99.0%

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

       17. unpow-1 99.0%

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

   13. Applied egg-rr 99.0%

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

       1. unpow2 99.0%

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

       2. sin-mult 98.9%

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

   15. Applied egg-rr 98.9%

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

       1. div-sub 98.9%

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

       2. +-inverses 98.9%

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

       3. cos-0 98.9%

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

       4. metadata-eval 98.9%

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

       5. count-2 98.9%

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

       6. *-commutative 98.9%

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

   17. Simplified 98.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000105:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot {k}^{-2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{k \cdot \frac{1}{\ell}}{\frac{1}{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (* (/ (* k_m (/ 1.0 l)) (cos k_m)) (* (pow (sin k_m) 2.0) (* t (/ k_m l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (((k_m * (1.0 / l)) / cos(k_m)) * (pow(sin(k_m), 2.0) * (t * (k_m / l))));
}

Fortran

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

Java

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

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (((k_m * (1.0 / l)) / math.cos(k_m)) * (math.pow(math.sin(k_m), 2.0) * (t * (k_m / l))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(Float64(k_m * Float64(1.0 / l)) / cos(k_m)) * Float64((sin(k_m) ^ 2.0) * Float64(t * Float64(k_m / l)))))
end

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(k$95$m * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate--l+ 33.9%

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

3. Simplified 33.9%

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

5. Taylor expanded in t around 0 70.8%

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

   1. times-frac 73.8%

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

7. Simplified 73.8%

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

   1. associate-*r/ 73.7%

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

   2. div-inv 73.4%

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

   3. pow-flip 73.4%

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

   4. metadata-eval 73.4%

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

9. Applied egg-rr 73.4%

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

    1. associate-/l* 73.4%

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

11. Simplified 73.4%

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

    1. add-sqr-sqrt 73.4%

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

    2. div-inv 73.4%

       \[\leadsto \frac{2}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}} \cdot \sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\color{blue}{\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}}}} \]

    3. times-frac 73.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}}} \]

    4. sqrt-prod 73.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    5. unpow2 73.4%

       \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    6. sqrt-prod 41.8%

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

    7. add-sqr-sqrt 56.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    8. sqrt-pow1 51.6%

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

    9. metadata-eval 51.6%

       \[\leadsto \frac{2}{\frac{k \cdot {\ell}^{\color{blue}{-1}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    10. unpow-1 51.6%

        \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\frac{1}{\ell}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    11. sqrt-prod 52.0%

        \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    12. unpow2 52.0%

        \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    13. sqrt-prod 28.5%

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

    14. add-sqr-sqrt 53.5%

        \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    15. sqrt-pow1 97.6%

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

    16. metadata-eval 97.6%

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

    17. unpow-1 97.6%

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

13. Applied egg-rr 97.6%

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

14. Taylor expanded in k around inf 93.4%

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

    1. associate-*l/ 97.7%

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

    2. *-commutative 97.7%

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

    3. *-commutative 97.7%

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

    4. associate-*l* 98.0%

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

16. Simplified 98.0%

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

17. Final simplification 98.0%

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

Alternative 3: 96.2% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (* (/ (/ k_m l) (cos k_m)) (* (/ k_m l) (* (pow (sin k_m) 2.0) t)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / l), $MachinePrecision] * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate--l+ 33.9%

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

3. Simplified 33.9%

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

5. Taylor expanded in t around 0 70.8%

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

   1. times-frac 73.8%

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

7. Simplified 73.8%

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

   1. associate-*r/ 73.7%

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

   2. div-inv 73.4%

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

   3. pow-flip 73.4%

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

   4. metadata-eval 73.4%

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

9. Applied egg-rr 73.4%

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

    1. associate-/l* 73.4%

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

11. Simplified 73.4%

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

    1. add-sqr-sqrt 73.4%

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

    2. pow2 73.4%

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

    3. sqrt-prod 73.4%

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

    4. unpow2 73.4%

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

    5. sqrt-prod 42.2%

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

    6. add-sqr-sqrt 76.5%

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

    7. sqrt-pow1 92.1%

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

    8. metadata-eval 92.1%

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

    9. unpow-1 92.1%

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

13. Applied egg-rr 92.1%

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

    1. unpow2 92.1%

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

    2. div-inv 92.2%

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

    3. frac-times 97.6%

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

    4. un-div-inv 97.6%

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

    5. div-inv 97.6%

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

    6. un-div-inv 97.6%

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

    7. inv-pow 97.6%

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

    8. pow-flip 97.6%

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

    9. metadata-eval 97.6%

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

    10. pow1 97.6%

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

15. Applied egg-rr 97.6%

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

16. Final simplification 97.6%

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

Alternative 4: 72.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := k_m \cdot \frac{1}{\ell}\\ \frac{2}{\frac{t_1}{\cos k_m} \cdot \frac{t_1}{\frac{1}{t \cdot {k_m}^{2}}}} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* k_m (/ 1.0 l))))
   (/ 2.0 (* (/ t_1 (cos k_m)) (/ t_1 (/ 1.0 (* t (pow k_m 2.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = k_m * (1.0 / l);
	return 2.0 / ((t_1 / cos(k_m)) * (t_1 / (1.0 / (t * pow(k_m, 2.0)))));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    t_1 = k_m * (1.0d0 / l)
    code = 2.0d0 / ((t_1 / cos(k_m)) * (t_1 / (1.0d0 / (t * (k_m ** 2.0d0)))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = k_m * (1.0 / l);
	return 2.0 / ((t_1 / Math.cos(k_m)) * (t_1 / (1.0 / (t * Math.pow(k_m, 2.0)))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = k_m * (1.0 / l)
	return 2.0 / ((t_1 / math.cos(k_m)) * (t_1 / (1.0 / (t * math.pow(k_m, 2.0)))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(k_m * Float64(1.0 / l))
	return Float64(2.0 / Float64(Float64(t_1 / cos(k_m)) * Float64(t_1 / Float64(1.0 / Float64(t * (k_m ^ 2.0))))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	t_1 = k_m * (1.0 / l);
	tmp = 2.0 / ((t_1 / cos(k_m)) * (t_1 / (1.0 / (t * (k_m ^ 2.0)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(k$95$m * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]}, N[(2.0 / N[(N[(t$95$1 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(1.0 / N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate--l+ 33.9%

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

3. Simplified 33.9%

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

5. Taylor expanded in t around 0 70.8%

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

   1. times-frac 73.8%

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

7. Simplified 73.8%

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

   1. associate-*r/ 73.7%

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

   2. div-inv 73.4%

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

   3. pow-flip 73.4%

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

   4. metadata-eval 73.4%

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

9. Applied egg-rr 73.4%

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

    1. associate-/l* 73.4%

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

11. Simplified 73.4%

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

    1. add-sqr-sqrt 73.4%

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

    2. div-inv 73.4%

       \[\leadsto \frac{2}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}} \cdot \sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\color{blue}{\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}}}} \]

    3. times-frac 73.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}}} \]

    4. sqrt-prod 73.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    5. unpow2 73.4%

       \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    6. sqrt-prod 41.8%

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

    7. add-sqr-sqrt 56.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    8. sqrt-pow1 51.6%

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

    9. metadata-eval 51.6%

       \[\leadsto \frac{2}{\frac{k \cdot {\ell}^{\color{blue}{-1}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    10. unpow-1 51.6%

        \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\frac{1}{\ell}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    11. sqrt-prod 52.0%

        \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    12. unpow2 52.0%

        \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    13. sqrt-prod 28.5%

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

    14. add-sqr-sqrt 53.5%

        \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

    15. sqrt-pow1 97.6%

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

    16. metadata-eval 97.6%

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

    17. unpow-1 97.6%

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

13. Applied egg-rr 97.6%

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

14. Taylor expanded in k around 0 70.8%

    \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{k \cdot \frac{1}{\ell}}{\frac{1}{\color{blue}{{k}^{2} \cdot t}}}} \]

15. Final simplification 70.8%

    \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{k \cdot \frac{1}{\ell}}{\frac{1}{t \cdot {k}^{2}}}} \]
16. Add Preprocessing

Alternative 5: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \ell \cdot {k_m}^{-2}\\ \mathbf{if}\;k_m \leq 1.15:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k_m \cdot \frac{1}{\ell}}{\cos k_m} \cdot \left(t \cdot \frac{{k_m}^{3}}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* l (pow k_m -2.0))))
   (if (<= k_m 1.15)
     (* 2.0 (* t_1 (/ t_1 t)))
     (/ 2.0 (* (/ (* k_m (/ 1.0 l)) (cos k_m)) (* t (/ (pow k_m 3.0) l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l * pow(k_m, -2.0);
	double tmp;
	if (k_m <= 1.15) {
		tmp = 2.0 * (t_1 * (t_1 / t));
	} else {
		tmp = 2.0 / (((k_m * (1.0 / l)) / cos(k_m)) * (t * (pow(k_m, 3.0) / l)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (k_m ** (-2.0d0))
    if (k_m <= 1.15d0) then
        tmp = 2.0d0 * (t_1 * (t_1 / t))
    else
        tmp = 2.0d0 / (((k_m * (1.0d0 / l)) / cos(k_m)) * (t * ((k_m ** 3.0d0) / l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l * Math.pow(k_m, -2.0);
	double tmp;
	if (k_m <= 1.15) {
		tmp = 2.0 * (t_1 * (t_1 / t));
	} else {
		tmp = 2.0 / (((k_m * (1.0 / l)) / Math.cos(k_m)) * (t * (Math.pow(k_m, 3.0) / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l * math.pow(k_m, -2.0)
	tmp = 0
	if k_m <= 1.15:
		tmp = 2.0 * (t_1 * (t_1 / t))
	else:
		tmp = 2.0 / (((k_m * (1.0 / l)) / math.cos(k_m)) * (t * (math.pow(k_m, 3.0) / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l * (k_m ^ -2.0))
	tmp = 0.0
	if (k_m <= 1.15)
		tmp = Float64(2.0 * Float64(t_1 * Float64(t_1 / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(1.0 / l)) / cos(k_m)) * Float64(t * Float64((k_m ^ 3.0) / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l * (k_m ^ -2.0);
	tmp = 0.0;
	if (k_m <= 1.15)
		tmp = 2.0 * (t_1 * (t_1 / t));
	else
		tmp = 2.0 / (((k_m * (1.0 / l)) / cos(k_m)) * (t * ((k_m ^ 3.0) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.15], N[(2.0 * N[(t$95$1 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[Power[k$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999

   1. Initial program 34.9%

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

      1. associate-*l* 35.0%

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

      2. associate-/r* 35.0%

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

      3. sub-neg 35.0%

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

      4. distribute-rgt-in 30.6%

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

      5. unpow2 30.6%

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

      6. times-frac 22.5%

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

      7. sqr-neg 22.5%

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

      8. times-frac 30.6%

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

      9. unpow2 30.6%

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

      10. distribute-rgt-in 35.0%

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

      11. +-commutative 35.0%

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

      12. associate-+l+ 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in k around 0 58.0%

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

      1. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

      1. expm1-log1p-u 57.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 55.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 55.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 57.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 57.6%

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

   11. Simplified 57.6%

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

       1. *-un-lft-identity 57.6%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{\color{blue}{1 \cdot t}} \]

       2. rem-cbrt-cube 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt[3]{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}}}}{1 \cdot t} \]

       3. unpow1/3 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{0.3333333333333333}}}{1 \cdot t} \]

       4. sqr-pow 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{1 \cdot t} \]

       5. times-frac 55.9%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right)} \]

       6. pow-pow 55.9%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \frac{0.3333333333333333}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       7. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \color{blue}{0.16666666666666666}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       8. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\color{blue}{0.5}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       9. pow1/2 55.9%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       10. sqrt-prod 55.9%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       11. unpow2 55.9%

           \[\leadsto 2 \cdot \left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       12. sqrt-prod 26.5%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       13. add-sqr-sqrt 37.7%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       14. sqrt-pow1 37.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       15. metadata-eval 37.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{\color{blue}{-2}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

   13. Applied egg-rr 74.4%

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

   if 1.1499999999999999 < k

   1. Initial program 31.2%

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

      1. associate-*l* 31.2%

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

      2. associate--l+ 31.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. times-frac 77.8%

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

   7. Simplified 77.8%

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

      1. associate-*r/ 77.7%

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

      2. div-inv 77.7%

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

      3. pow-flip 77.7%

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

      4. metadata-eval 77.7%

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

   9. Applied egg-rr 77.7%

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

       1. associate-/l* 77.7%

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

   11. Simplified 77.7%

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

       1. add-sqr-sqrt 77.7%

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

       2. div-inv 77.8%

          \[\leadsto \frac{2}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}} \cdot \sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\color{blue}{\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}}}} \]

       3. times-frac 77.7%

          \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}}} \]

       4. sqrt-prod 77.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       5. unpow2 77.7%

          \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       6. sqrt-prod 77.8%

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

       7. add-sqr-sqrt 77.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       8. sqrt-pow1 66.6%

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

       9. metadata-eval 66.6%

          \[\leadsto \frac{2}{\frac{k \cdot {\ell}^{\color{blue}{-1}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       10. unpow-1 66.6%

           \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\frac{1}{\ell}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       11. sqrt-prod 67.8%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       12. unpow2 67.8%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       13. sqrt-prod 69.3%

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

       14. add-sqr-sqrt 69.4%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       15. sqrt-pow1 99.0%

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

       16. metadata-eval 99.0%

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

       17. unpow-1 99.0%

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

   13. Applied egg-rr 99.0%

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

   14. Taylor expanded in k around 0 62.4%

       \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \color{blue}{\frac{{k}^{3} \cdot t}{\ell}}} \]
   15. Step-by-step derivation

       1. associate-/l* 64.8%

          \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \color{blue}{\frac{{k}^{3}}{\frac{\ell}{t}}}} \]

       2. associate-/r/ 62.4%

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

   16. Simplified 62.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot {k}^{-2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \left(t \cdot \frac{{k}^{3}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \ell \cdot {k_m}^{-2}\\ \mathbf{if}\;k_m \leq 1.25:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k_m \cdot \frac{1}{\ell}}{\cos k_m} \cdot \frac{{k_m}^{3}}{\frac{\ell}{t}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* l (pow k_m -2.0))))
   (if (<= k_m 1.25)
     (* 2.0 (* t_1 (/ t_1 t)))
     (/ 2.0 (* (/ (* k_m (/ 1.0 l)) (cos k_m)) (/ (pow k_m 3.0) (/ l t)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l * pow(k_m, -2.0);
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 * (t_1 * (t_1 / t));
	} else {
		tmp = 2.0 / (((k_m * (1.0 / l)) / cos(k_m)) * (pow(k_m, 3.0) / (l / t)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l * (k_m ** (-2.0d0))
    if (k_m <= 1.25d0) then
        tmp = 2.0d0 * (t_1 * (t_1 / t))
    else
        tmp = 2.0d0 / (((k_m * (1.0d0 / l)) / cos(k_m)) * ((k_m ** 3.0d0) / (l / t)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l * Math.pow(k_m, -2.0);
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 * (t_1 * (t_1 / t));
	} else {
		tmp = 2.0 / (((k_m * (1.0 / l)) / Math.cos(k_m)) * (Math.pow(k_m, 3.0) / (l / t)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l * math.pow(k_m, -2.0)
	tmp = 0
	if k_m <= 1.25:
		tmp = 2.0 * (t_1 * (t_1 / t))
	else:
		tmp = 2.0 / (((k_m * (1.0 / l)) / math.cos(k_m)) * (math.pow(k_m, 3.0) / (l / t)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l * (k_m ^ -2.0))
	tmp = 0.0
	if (k_m <= 1.25)
		tmp = Float64(2.0 * Float64(t_1 * Float64(t_1 / t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(1.0 / l)) / cos(k_m)) * Float64((k_m ^ 3.0) / Float64(l / t))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = l * (k_m ^ -2.0);
	tmp = 0.0;
	if (k_m <= 1.25)
		tmp = 2.0 * (t_1 * (t_1 / t));
	else
		tmp = 2.0 / (((k_m * (1.0 / l)) / cos(k_m)) * ((k_m ^ 3.0) / (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k$95$m, 1.25], N[(2.0 * N[(t$95$1 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 3.0], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25

   1. Initial program 34.9%

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

      1. associate-*l* 35.0%

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

      2. associate-/r* 35.0%

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

      3. sub-neg 35.0%

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

      4. distribute-rgt-in 30.6%

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

      5. unpow2 30.6%

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

      6. times-frac 22.5%

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

      7. sqr-neg 22.5%

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

      8. times-frac 30.6%

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

      9. unpow2 30.6%

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

      10. distribute-rgt-in 35.0%

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

      11. +-commutative 35.0%

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

      12. associate-+l+ 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in k around 0 58.0%

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

      1. associate-/r* 57.6%

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

   7. Simplified 57.6%

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

      1. expm1-log1p-u 57.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

      2. expm1-udef 55.5%

         \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

      3. div-inv 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

      4. pow-flip 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

      5. metadata-eval 55.5%

         \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

   9. Applied egg-rr 55.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
   10. Step-by-step derivation

       1. expm1-def 57.5%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

       2. expm1-log1p 57.6%

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

   11. Simplified 57.6%

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

       1. *-un-lft-identity 57.6%

          \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{\color{blue}{1 \cdot t}} \]

       2. rem-cbrt-cube 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt[3]{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}}}}{1 \cdot t} \]

       3. unpow1/3 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{0.3333333333333333}}}{1 \cdot t} \]

       4. sqr-pow 55.9%

          \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{1 \cdot t} \]

       5. times-frac 55.9%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right)} \]

       6. pow-pow 55.9%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \frac{0.3333333333333333}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       7. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \color{blue}{0.16666666666666666}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       8. metadata-eval 55.9%

          \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\color{blue}{0.5}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       9. pow1/2 55.9%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       10. sqrt-prod 55.9%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       11. unpow2 55.9%

           \[\leadsto 2 \cdot \left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       12. sqrt-prod 26.5%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       13. add-sqr-sqrt 37.7%

           \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       14. sqrt-pow1 37.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

       15. metadata-eval 37.7%

           \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{\color{blue}{-2}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

   13. Applied egg-rr 74.4%

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

   if 1.25 < k

   1. Initial program 31.2%

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

      1. associate-*l* 31.2%

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

      2. associate--l+ 31.2%

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

   3. Simplified 31.2%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. times-frac 77.8%

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

   7. Simplified 77.8%

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

      1. associate-*r/ 77.7%

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

      2. div-inv 77.7%

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

      3. pow-flip 77.7%

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

      4. metadata-eval 77.7%

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

   9. Applied egg-rr 77.7%

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

       1. associate-/l* 77.7%

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

   11. Simplified 77.7%

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

       1. add-sqr-sqrt 77.7%

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

       2. div-inv 77.8%

          \[\leadsto \frac{2}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}} \cdot \sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\color{blue}{\cos k \cdot \frac{1}{t \cdot {\sin k}^{2}}}}} \]

       3. times-frac 77.7%

          \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}}} \]

       4. sqrt-prod 77.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       5. unpow2 77.7%

          \[\leadsto \frac{2}{\frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       6. sqrt-prod 77.8%

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

       7. add-sqr-sqrt 77.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       8. sqrt-pow1 66.6%

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

       9. metadata-eval 66.6%

          \[\leadsto \frac{2}{\frac{k \cdot {\ell}^{\color{blue}{-1}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       10. unpow-1 66.6%

           \[\leadsto \frac{2}{\frac{k \cdot \color{blue}{\frac{1}{\ell}}}{\cos k} \cdot \frac{\sqrt{{k}^{2} \cdot {\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       11. sqrt-prod 67.8%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{\ell}^{-2}}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       12. unpow2 67.8%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\sqrt{\color{blue}{k \cdot k}} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       13. sqrt-prod 69.3%

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

       14. add-sqr-sqrt 69.4%

           \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{\color{blue}{k} \cdot \sqrt{{\ell}^{-2}}}{\frac{1}{t \cdot {\sin k}^{2}}}} \]

       15. sqrt-pow1 99.0%

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

       16. metadata-eval 99.0%

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

       17. unpow-1 99.0%

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

   13. Applied egg-rr 99.0%

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

   14. Taylor expanded in k around 0 62.4%

       \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \color{blue}{\frac{{k}^{3} \cdot t}{\ell}}} \]
   15. Step-by-step derivation

       1. associate-/l* 64.8%

          \[\leadsto \frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \color{blue}{\frac{{k}^{3}}{\frac{\ell}{t}}}} \]

   16. Simplified 64.8%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot {k}^{-2}\right) \cdot \frac{\ell \cdot {k}^{-2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{1}{\ell}}{\cos k} \cdot \frac{{k}^{3}}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \ell \cdot {k_m}^{-2}\\ 2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right) \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* l (pow k_m -2.0)))) (* 2.0 (* t_1 (/ t_1 t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = l * pow(k_m, -2.0);
	return 2.0 * (t_1 * (t_1 / t));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    t_1 = l * (k_m ** (-2.0d0))
    code = 2.0d0 * (t_1 * (t_1 / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = l * Math.pow(k_m, -2.0);
	return 2.0 * (t_1 * (t_1 / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = l * math.pow(k_m, -2.0)
	return 2.0 * (t_1 * (t_1 / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(l * (k_m ^ -2.0))
	return Float64(2.0 * Float64(t_1 * Float64(t_1 / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	t_1 = l * (k_m ^ -2.0);
	tmp = 2.0 * (t_1 * (t_1 / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(t$95$1 * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate-/r* 33.9%

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

   3. sub-neg 33.9%

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

   4. distribute-rgt-in 30.8%

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

   5. unpow2 30.8%

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

   6. times-frac 22.6%

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

   7. sqr-neg 22.6%

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

   8. times-frac 30.8%

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

   9. unpow2 30.8%

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

   10. distribute-rgt-in 33.9%

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

   11. +-commutative 33.9%

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

   12. associate-+l+ 39.4%

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

3. Simplified 39.4%

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

5. Taylor expanded in k around 0 57.4%

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

   1. associate-/r* 57.1%

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

7. Simplified 57.1%

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

   1. expm1-log1p-u 57.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 55.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 55.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 55.7%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 57.1%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 57.2%

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

11. Simplified 57.2%

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

    1. *-un-lft-identity 57.2%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{\color{blue}{1 \cdot t}} \]

    2. rem-cbrt-cube 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt[3]{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}}}}{1 \cdot t} \]

    3. unpow1/3 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{0.3333333333333333}}}{1 \cdot t} \]

    4. sqr-pow 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{1 \cdot t} \]

    5. times-frac 56.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right)} \]

    6. pow-pow 56.0%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \frac{0.3333333333333333}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    7. metadata-eval 56.0%

       \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \color{blue}{0.16666666666666666}\right)}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    8. metadata-eval 56.0%

       \[\leadsto 2 \cdot \left(\frac{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\color{blue}{0.5}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    9. pow1/2 56.0%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2} \cdot {k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    10. sqrt-prod 56.0%

        \[\leadsto 2 \cdot \left(\frac{\color{blue}{\sqrt{{\ell}^{2}} \cdot \sqrt{{k}^{-4}}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    11. unpow2 56.0%

        \[\leadsto 2 \cdot \left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    12. sqrt-prod 25.5%

        \[\leadsto 2 \cdot \left(\frac{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    13. add-sqr-sqrt 43.6%

        \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell} \cdot \sqrt{{k}^{-4}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    14. sqrt-pow1 43.6%

        \[\leadsto 2 \cdot \left(\frac{\ell \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

    15. metadata-eval 43.6%

        \[\leadsto 2 \cdot \left(\frac{\ell \cdot {k}^{\color{blue}{-2}}}{1} \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{t}\right) \]

13. Applied egg-rr 69.5%

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

14. Final simplification 69.5%

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

Alternative 8: 59.9% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (* (pow k_m -4.0) (/ (pow l 2.0) t))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(N[Power[k$95$m, -4.0], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate-/r* 33.9%

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

   3. sub-neg 33.9%

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

   4. distribute-rgt-in 30.8%

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

   5. unpow2 30.8%

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

   6. times-frac 22.6%

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

   7. sqr-neg 22.6%

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

   8. times-frac 30.8%

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

   9. unpow2 30.8%

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

   10. distribute-rgt-in 33.9%

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

   11. +-commutative 33.9%

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

   12. associate-+l+ 39.4%

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

3. Simplified 39.4%

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

5. Taylor expanded in k around 0 57.4%

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

   1. associate-/r* 57.1%

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

7. Simplified 57.1%

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

   1. expm1-log1p-u 57.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 55.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 55.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 55.7%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 57.1%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 57.2%

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

11. Simplified 57.2%

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

    1. expm1-log1p-u 40.5%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot {k}^{-4}}{t}\right)\right)} \]

    2. expm1-udef 39.5%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot {k}^{-4}}{t}\right)} - 1\right)} \]

    3. associate-/l* 39.8%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{\frac{t}{{k}^{-4}}}}\right)} - 1\right) \]

13. Applied egg-rr 39.8%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{\frac{t}{{k}^{-4}}}\right)} - 1\right)} \]
14. Step-by-step derivation

    1. expm1-def 40.8%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{\frac{t}{{k}^{-4}}}\right)\right)} \]

    2. expm1-log1p 57.5%

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

    3. associate-/r/ 56.7%

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

    4. *-commutative 56.7%

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

15. Simplified 56.7%

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

16. Final simplification 56.7%

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

Alternative 9: 60.7% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (* (pow l 2.0) (/ (pow k_m -4.0) t))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[k$95$m, -4.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate-/r* 33.9%

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

   3. sub-neg 33.9%

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

   4. distribute-rgt-in 30.8%

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

   5. unpow2 30.8%

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

   6. times-frac 22.6%

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

   7. sqr-neg 22.6%

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

   8. times-frac 30.8%

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

   9. unpow2 30.8%

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

   10. distribute-rgt-in 33.9%

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

   11. +-commutative 33.9%

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

   12. associate-+l+ 39.4%

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

3. Simplified 39.4%

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

5. Taylor expanded in k around 0 57.4%

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

   1. associate-/r* 57.1%

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

7. Simplified 57.1%

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

   1. expm1-log1p-u 57.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 55.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 55.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 55.7%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 57.1%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 57.2%

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

11. Simplified 57.2%

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

    1. expm1-log1p-u 40.5%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot {k}^{-4}}{t}\right)\right)} \]

    2. expm1-udef 39.5%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2} \cdot {k}^{-4}}{t}\right)} - 1\right)} \]

    3. associate-/l* 39.8%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{\frac{t}{{k}^{-4}}}}\right)} - 1\right) \]

13. Applied egg-rr 39.8%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{\frac{t}{{k}^{-4}}}\right)} - 1\right)} \]
14. Step-by-step derivation

    1. expm1-def 40.8%

       \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{\frac{t}{{k}^{-4}}}\right)\right)} \]

    2. expm1-log1p 57.5%

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

    3. associate-/l* 57.2%

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

    4. associate-*r/ 57.5%

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

15. Simplified 57.5%

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

16. Final simplification 57.5%

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

Alternative 10: 70.6% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (/ (pow (* l (pow k_m -2.0)) 2.0) t)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(N[Power[N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

   1. associate-*l* 33.9%

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

   2. associate-/r* 33.9%

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

   3. sub-neg 33.9%

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

   4. distribute-rgt-in 30.8%

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

   5. unpow2 30.8%

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

   6. times-frac 22.6%

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

   7. sqr-neg 22.6%

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

   8. times-frac 30.8%

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

   9. unpow2 30.8%

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

   10. distribute-rgt-in 33.9%

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

   11. +-commutative 33.9%

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

   12. associate-+l+ 39.4%

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

3. Simplified 39.4%

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

5. Taylor expanded in k around 0 57.4%

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

   1. associate-/r* 57.1%

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

7. Simplified 57.1%

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

   1. expm1-log1p-u 57.1%

      \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

   2. expm1-udef 55.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

   3. div-inv 55.6%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{{k}^{4}}}\right)} - 1}{t} \]

   4. pow-flip 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1}{t} \]

   5. metadata-eval 55.7%

      \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{\color{blue}{-4}}\right)} - 1}{t} \]

9. Applied egg-rr 55.7%

   \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)} - 1}}{t} \]
10. Step-by-step derivation

    1. expm1-def 57.1%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{2} \cdot {k}^{-4}\right)\right)}}{t} \]

    2. expm1-log1p 57.2%

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

11. Simplified 57.2%

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

    1. rem-cbrt-cube 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt[3]{{\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}}}}{t} \]

    2. unpow1/3 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{0.3333333333333333}}}{t} \]

    3. sqr-pow 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{t} \]

    4. pow2 56.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{{\left({\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}^{2}}}{t} \]

    5. pow-pow 57.2%

       \[\leadsto 2 \cdot \frac{{\color{blue}{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \frac{0.3333333333333333}{2}\right)}\right)}}^{2}}{t} \]

    6. metadata-eval 57.2%

       \[\leadsto 2 \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\left(3 \cdot \color{blue}{0.16666666666666666}\right)}\right)}^{2}}{t} \]

    7. metadata-eval 57.2%

       \[\leadsto 2 \cdot \frac{{\left({\left({\ell}^{2} \cdot {k}^{-4}\right)}^{\color{blue}{0.5}}\right)}^{2}}{t} \]

    8. pow1/2 57.2%

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

    9. sqrt-prod 57.2%

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

    10. unpow2 57.2%

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

    11. sqrt-prod 30.3%

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

    12. add-sqr-sqrt 63.8%

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

    13. sqrt-pow1 67.6%

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

    14. metadata-eval 67.6%

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

13. Applied egg-rr 67.6%

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

14. Final simplification 67.6%

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

Reproduce

herbie shell --seed 2024020 
(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))))