Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.5% → 92.9%

Time: 19.1s

Alternatives: 15

Speedup: 3.8×

• 39.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.5% 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: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\tan k\_m \cdot \sin k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.000305:\\ \;\;\;\;{\left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k\_m}^{2}}{t\_m}, \frac{2}{t\_m}\right)} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{{\left(\sqrt{\sqrt[3]{l\_m}}\right)}^{-4}}}{t\_2}\right)}^{2}}{t\_m \cdot \left({\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (tan k_m) (sin k_m)))))
   (*
    t_s
    (if (<= k_m 0.000305)
      (pow
       (*
        l_m
        (*
         (sqrt (fma -0.3333333333333333 (/ (pow k_m 2.0) t_m) (/ 2.0 t_m)))
         (pow k_m -2.0)))
       2.0)
      (/
       (pow (/ (/ (/ (sqrt 2.0) k_m) (pow (sqrt (cbrt l_m)) -4.0)) t_2) 2.0)
       (* t_m (* (pow (cbrt l_m) -2.0) t_2)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt((tan(k_m) * sin(k_m)));
	double tmp;
	if (k_m <= 0.000305) {
		tmp = pow((l_m * (sqrt(fma(-0.3333333333333333, (pow(k_m, 2.0) / t_m), (2.0 / t_m))) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = pow((((sqrt(2.0) / k_m) / pow(sqrt(cbrt(l_m)), -4.0)) / t_2), 2.0) / (t_m * (pow(cbrt(l_m), -2.0) * t_2));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(Float64(tan(k_m) * sin(k_m)))
	tmp = 0.0
	if (k_m <= 0.000305)
		tmp = Float64(l_m * Float64(sqrt(fma(-0.3333333333333333, Float64((k_m ^ 2.0) / t_m), Float64(2.0 / t_m))) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / (sqrt(cbrt(l_m)) ^ -4.0)) / t_2) ^ 2.0) / Float64(t_m * Float64((cbrt(l_m) ^ -2.0) * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.000305], N[Power[N[(l$95$m * N[(N[Sqrt[N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / N[Power[N[Sqrt[N[Power[l$95$m, 1/3], $MachinePrecision]], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.04999999999999987e-4

   1. Initial program 35.4%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 27.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-/l* 26.6%

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

      4. associate-*r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. associate-/l* 26.6%

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

   7. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. associate-*r/ 27.7%

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

      4. pow2 27.7%

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

      5. pow-prod-down 47.5%

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

      6. pow2 47.5%

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

      7. *-commutative 47.5%

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

      8. pow2 47.5%

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

      9. pow-flip 47.5%

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

      10. metadata-eval 47.5%

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

   9. Applied egg-rr 47.5%

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

       1. *-commutative 47.5%

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

       2. fma-define 47.5%

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

       3. associate-*r/ 47.5%

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

       4. unpow2 47.5%

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

       5. swap-sqr 27.7%

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

       6. unpow2 27.7%

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

       7. unpow2 27.7%

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

       8. *-commutative 27.7%

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

       9. associate-*r* 27.7%

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

       10. associate-*l/ 26.6%

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

       11. associate-*r/ 26.6%

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

       12. distribute-rgt-in 43.8%

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

       13. metadata-eval 43.8%

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

       14. associate-*r/ 43.8%

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

       15. fma-define 43.8%

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

   11. Simplified 43.8%

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

       1. add-sqr-sqrt 24.4%

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

       2. pow2 24.4%

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

       3. *-commutative 24.4%

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

       4. pow2 24.4%

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

       5. sqrt-prod 24.4%

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

       6. sqrt-prod 22.3%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 28.8%

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

       9. sqrt-pow1 34.4%

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

       10. metadata-eval 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-*l* 35.0%

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

   15. Simplified 35.0%

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

   if 3.04999999999999987e-4 < k

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r* 32.2%

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

   3. Simplified 38.1%

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

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.0%

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

      3. times-frac 38.0%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r/ 89.1%

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

      2. associate-/r* 89.1%

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

      3. associate-/r/ 89.1%

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

   8. Simplified 89.1%

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

      1. frac-times 81.9%

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

      2. associate-/l* 81.9%

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

      3. div-inv 81.8%

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

      4. pow-flip 81.8%

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

      5. metadata-eval 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. times-frac 88.9%

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

       2. associate-*r/ 89.0%

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

       3. associate-/r* 88.9%

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

       4. associate-*l/ 81.8%

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

       5. unpow2 81.8%

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

   12. Simplified 94.5%

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

       1. add-sqr-sqrt 44.7%

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

       2. unpow-prod-down 44.7%

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

   14. Applied egg-rr 44.7%

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

       1. pow-sqr 44.7%

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

       2. metadata-eval 44.7%

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

   16. Simplified 44.7%

       \[\leadsto \frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k}}{\color{blue}{{\left(\sqrt{\sqrt[3]{\ell}}\right)}^{-4}}}}{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.000305:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2}}{t}, \frac{2}{t}\right)} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt{\sqrt[3]{\ell}}\right)}^{-4}}}{\sqrt[3]{\tan k \cdot \sin k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\tan k \cdot \sin k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\\ t_3 := {\left(\sqrt[3]{l\_m}\right)}^{-2}\\ t\_s \cdot \frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{t\_3}}{t\_2}\right)}^{2}}{t\_m \cdot \left(t\_3 \cdot t\_2\right)} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (cbrt (tan k_m)) (cbrt (sin k_m))))
        (t_3 (pow (cbrt l_m) -2.0)))
   (*
    t_s
    (/ (pow (/ (/ (/ (sqrt 2.0) k_m) t_3) t_2) 2.0) (* t_m (* t_3 t_2))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt(tan(k_m)) * cbrt(sin(k_m));
	double t_3 = pow(cbrt(l_m), -2.0);
	return t_s * (pow((((sqrt(2.0) / k_m) / t_3) / t_2), 2.0) / (t_m * (t_3 * t_2)));
}

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m));
	double t_3 = Math.pow(Math.cbrt(l_m), -2.0);
	return t_s * (Math.pow((((Math.sqrt(2.0) / k_m) / t_3) / t_2), 2.0) / (t_m * (t_3 * t_2)));
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(cbrt(tan(k_m)) * cbrt(sin(k_m)))
	t_3 = cbrt(l_m) ^ -2.0
	return Float64(t_s * Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / t_3) / t_2) ^ 2.0) / Float64(t_m * Float64(t_3 * t_2))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 34.5%

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

   1. *-commutative 34.5%

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

   2. associate-/r* 34.6%

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

3. Simplified 40.4%

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

   1. add-sqr-sqrt 40.4%

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

   2. add-cube-cbrt 40.4%

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

   3. times-frac 40.4%

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

6. Applied egg-rr 82.8%

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

   1. associate-/r/ 82.8%

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

   2. associate-/r* 82.9%

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

   3. associate-/r/ 83.1%

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

8. Simplified 83.1%

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

   1. frac-times 79.2%

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

   2. associate-/l* 79.2%

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

   3. div-inv 79.2%

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

   4. pow-flip 79.1%

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

   5. metadata-eval 79.1%

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

10. Applied egg-rr 79.1%

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

    1. times-frac 83.0%

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

    2. associate-*r/ 83.0%

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

    3. associate-/r* 83.1%

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

    4. associate-*l/ 79.1%

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

    5. unpow2 79.1%

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

12. Simplified 92.5%

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

    1. *-commutative 92.5%

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

    2. cbrt-prod 92.5%

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

14. Applied egg-rr 92.5%

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

    1. *-commutative 92.5%

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

    2. cbrt-prod 92.5%

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

16. Applied egg-rr 93.2%

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

Alternative 3: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{l\_m}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0011:\\ \;\;\;\;{\left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k\_m}^{2}}{t\_m}, \frac{2}{t\_m}\right)} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{t\_2}}{\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}}\right)}^{2}}{t\_m \cdot \left(t\_2 \cdot \sqrt[3]{\tan k\_m \cdot \sin k\_m}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l_m) -2.0)))
   (*
    t_s
    (if (<= k_m 0.0011)
      (pow
       (*
        l_m
        (*
         (sqrt (fma -0.3333333333333333 (/ (pow k_m 2.0) t_m) (/ 2.0 t_m)))
         (pow k_m -2.0)))
       2.0)
      (/
       (pow
        (/ (/ (/ (sqrt 2.0) k_m) t_2) (* (cbrt (tan k_m)) (cbrt (sin k_m))))
        2.0)
       (* t_m (* t_2 (cbrt (* (tan k_m) (sin k_m))))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = pow(cbrt(l_m), -2.0);
	double tmp;
	if (k_m <= 0.0011) {
		tmp = pow((l_m * (sqrt(fma(-0.3333333333333333, (pow(k_m, 2.0) / t_m), (2.0 / t_m))) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = pow((((sqrt(2.0) / k_m) / t_2) / (cbrt(tan(k_m)) * cbrt(sin(k_m)))), 2.0) / (t_m * (t_2 * cbrt((tan(k_m) * sin(k_m)))));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(l_m) ^ -2.0
	tmp = 0.0
	if (k_m <= 0.0011)
		tmp = Float64(l_m * Float64(sqrt(fma(-0.3333333333333333, Float64((k_m ^ 2.0) / t_m), Float64(2.0 / t_m))) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / t_2) / Float64(cbrt(tan(k_m)) * cbrt(sin(k_m)))) ^ 2.0) / Float64(t_m * Float64(t_2 * cbrt(Float64(tan(k_m) * sin(k_m))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.0011], N[Power[N[(l$95$m * N[(N[Sqrt[N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(t$95$2 * N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00110000000000000007

   1. Initial program 35.4%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 27.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-/l* 26.6%

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

      4. associate-*r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. associate-/l* 26.6%

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

   7. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. associate-*r/ 27.7%

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

      4. pow2 27.7%

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

      5. pow-prod-down 47.5%

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

      6. pow2 47.5%

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

      7. *-commutative 47.5%

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

      8. pow2 47.5%

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

      9. pow-flip 47.5%

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

      10. metadata-eval 47.5%

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

   9. Applied egg-rr 47.5%

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

       1. *-commutative 47.5%

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

       2. fma-define 47.5%

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

       3. associate-*r/ 47.5%

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

       4. unpow2 47.5%

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

       5. swap-sqr 27.7%

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

       6. unpow2 27.7%

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

       7. unpow2 27.7%

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

       8. *-commutative 27.7%

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

       9. associate-*r* 27.7%

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

       10. associate-*l/ 26.6%

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

       11. associate-*r/ 26.6%

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

       12. distribute-rgt-in 43.8%

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

       13. metadata-eval 43.8%

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

       14. associate-*r/ 43.8%

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

       15. fma-define 43.8%

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

   11. Simplified 43.8%

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

       1. add-sqr-sqrt 24.4%

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

       2. pow2 24.4%

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

       3. *-commutative 24.4%

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

       4. pow2 24.4%

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

       5. sqrt-prod 24.4%

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

       6. sqrt-prod 22.3%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 28.8%

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

       9. sqrt-pow1 34.4%

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

       10. metadata-eval 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-*l* 35.0%

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

   15. Simplified 35.0%

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

   if 0.00110000000000000007 < k

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r* 32.2%

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

   3. Simplified 38.1%

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

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.0%

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

      3. times-frac 38.0%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r/ 89.1%

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

      2. associate-/r* 89.1%

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

      3. associate-/r/ 89.1%

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

   8. Simplified 89.1%

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

      1. frac-times 81.9%

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

      2. associate-/l* 81.9%

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

      3. div-inv 81.8%

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

      4. pow-flip 81.8%

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

      5. metadata-eval 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. times-frac 88.9%

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

       2. associate-*r/ 89.0%

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

       3. associate-/r* 88.9%

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

       4. associate-*l/ 81.8%

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

       5. unpow2 81.8%

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

   12. Simplified 94.5%

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

       1. *-commutative 94.5%

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

       2. cbrt-prod 94.5%

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

   14. Applied egg-rr 94.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0011:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2}}{t}, \frac{2}{t}\right)} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\tan k \cdot \sin k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\tan k\_m \cdot \sin k\_m}\\ t_3 := {\left(\sqrt[3]{l\_m}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00051:\\ \;\;\;\;{\left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k\_m}^{2}}{t\_m}, \frac{2}{t\_m}\right)} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k\_m}}{t\_3}}{t\_2}\right)}^{2}}{t\_m \cdot \left(t\_3 \cdot t\_2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (tan k_m) (sin k_m)))) (t_3 (pow (cbrt l_m) -2.0)))
   (*
    t_s
    (if (<= k_m 0.00051)
      (pow
       (*
        l_m
        (*
         (sqrt (fma -0.3333333333333333 (/ (pow k_m 2.0) t_m) (/ 2.0 t_m)))
         (pow k_m -2.0)))
       2.0)
      (/ (pow (/ (/ (/ (sqrt 2.0) k_m) t_3) t_2) 2.0) (* t_m (* t_3 t_2)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt((tan(k_m) * sin(k_m)));
	double t_3 = pow(cbrt(l_m), -2.0);
	double tmp;
	if (k_m <= 0.00051) {
		tmp = pow((l_m * (sqrt(fma(-0.3333333333333333, (pow(k_m, 2.0) / t_m), (2.0 / t_m))) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = pow((((sqrt(2.0) / k_m) / t_3) / t_2), 2.0) / (t_m * (t_3 * t_2));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(Float64(tan(k_m) * sin(k_m)))
	t_3 = cbrt(l_m) ^ -2.0
	tmp = 0.0
	if (k_m <= 0.00051)
		tmp = Float64(l_m * Float64(sqrt(fma(-0.3333333333333333, Float64((k_m ^ 2.0) / t_m), Float64(2.0 / t_m))) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(Float64(sqrt(2.0) / k_m) / t_3) / t_2) ^ 2.0) / Float64(t_m * Float64(t_3 * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.00051], N[Power[N[(l$95$m * N[(N[Sqrt[N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.1e-4

   1. Initial program 35.4%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 27.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-/l* 26.6%

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

      4. associate-*r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. associate-/l* 26.6%

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

   7. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. associate-*r/ 27.7%

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

      4. pow2 27.7%

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

      5. pow-prod-down 47.5%

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

      6. pow2 47.5%

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

      7. *-commutative 47.5%

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

      8. pow2 47.5%

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

      9. pow-flip 47.5%

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

      10. metadata-eval 47.5%

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

   9. Applied egg-rr 47.5%

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

       1. *-commutative 47.5%

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

       2. fma-define 47.5%

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

       3. associate-*r/ 47.5%

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

       4. unpow2 47.5%

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

       5. swap-sqr 27.7%

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

       6. unpow2 27.7%

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

       7. unpow2 27.7%

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

       8. *-commutative 27.7%

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

       9. associate-*r* 27.7%

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

       10. associate-*l/ 26.6%

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

       11. associate-*r/ 26.6%

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

       12. distribute-rgt-in 43.8%

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

       13. metadata-eval 43.8%

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

       14. associate-*r/ 43.8%

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

       15. fma-define 43.8%

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

   11. Simplified 43.8%

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

       1. add-sqr-sqrt 24.4%

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

       2. pow2 24.4%

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

       3. *-commutative 24.4%

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

       4. pow2 24.4%

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

       5. sqrt-prod 24.4%

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

       6. sqrt-prod 22.3%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 28.8%

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

       9. sqrt-pow1 34.4%

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

       10. metadata-eval 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-*l* 35.0%

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

   15. Simplified 35.0%

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

   if 5.1e-4 < k

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r* 32.2%

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

   3. Simplified 38.1%

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

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.0%

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

      3. times-frac 38.0%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r/ 89.1%

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

      2. associate-/r* 89.1%

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

      3. associate-/r/ 89.1%

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

   8. Simplified 89.1%

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

      1. frac-times 81.9%

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

      2. associate-/l* 81.9%

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

      3. div-inv 81.8%

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

      4. pow-flip 81.8%

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

      5. metadata-eval 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. times-frac 88.9%

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

       2. associate-*r/ 89.0%

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

       3. associate-/r* 88.9%

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

       4. associate-*l/ 81.8%

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

       5. unpow2 81.8%

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

   12. Simplified 94.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00051:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2}}{t}, \frac{2}{t}\right)} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\tan k \cdot \sin k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\tan k \cdot \sin k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot \sqrt[3]{\tan k\_m \cdot \sin k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00122:\\ \;\;\;\;{\left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k\_m}^{2}}{t\_m}, \frac{2}{t\_m}\right)} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\sqrt{2}}{k\_m}}{t\_2}\right)}^{2}}{t\_m \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (let* ((t_2 (* (pow (cbrt l_m) -2.0) (cbrt (* (tan k_m) (sin k_m))))))
   (*
    t_s
    (if (<= k_m 0.00122)
      (pow
       (*
        l_m
        (*
         (sqrt (fma -0.3333333333333333 (/ (pow k_m 2.0) t_m) (/ 2.0 t_m)))
         (pow k_m -2.0)))
       2.0)
      (/ (pow (/ (/ (sqrt 2.0) k_m) t_2) 2.0) (* t_m t_2))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = pow(cbrt(l_m), -2.0) * cbrt((tan(k_m) * sin(k_m)));
	double tmp;
	if (k_m <= 0.00122) {
		tmp = pow((l_m * (sqrt(fma(-0.3333333333333333, (pow(k_m, 2.0) / t_m), (2.0 / t_m))) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = pow(((sqrt(2.0) / k_m) / t_2), 2.0) / (t_m * t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64((cbrt(l_m) ^ -2.0) * cbrt(Float64(tan(k_m) * sin(k_m))))
	tmp = 0.0
	if (k_m <= 0.00122)
		tmp = Float64(l_m * Float64(sqrt(fma(-0.3333333333333333, Float64((k_m ^ 2.0) / t_m), Float64(2.0 / t_m))) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = Float64((Float64(Float64(sqrt(2.0) / k_m) / t_2) ^ 2.0) / Float64(t_m * t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := Block[{t$95$2 = N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.00122], N[Power[N[(l$95$m * N[(N[Sqrt[N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.00121999999999999995

   1. Initial program 35.4%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 27.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-/l* 26.6%

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

      4. associate-*r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. associate-/l* 26.6%

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

   7. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. associate-*r/ 27.7%

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

      4. pow2 27.7%

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

      5. pow-prod-down 47.5%

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

      6. pow2 47.5%

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

      7. *-commutative 47.5%

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

      8. pow2 47.5%

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

      9. pow-flip 47.5%

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

      10. metadata-eval 47.5%

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

   9. Applied egg-rr 47.5%

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

       1. *-commutative 47.5%

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

       2. fma-define 47.5%

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

       3. associate-*r/ 47.5%

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

       4. unpow2 47.5%

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

       5. swap-sqr 27.7%

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

       6. unpow2 27.7%

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

       7. unpow2 27.7%

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

       8. *-commutative 27.7%

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

       9. associate-*r* 27.7%

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

       10. associate-*l/ 26.6%

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

       11. associate-*r/ 26.6%

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

       12. distribute-rgt-in 43.8%

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

       13. metadata-eval 43.8%

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

       14. associate-*r/ 43.8%

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

       15. fma-define 43.8%

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

   11. Simplified 43.8%

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

       1. add-sqr-sqrt 24.4%

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

       2. pow2 24.4%

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

       3. *-commutative 24.4%

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

       4. pow2 24.4%

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

       5. sqrt-prod 24.4%

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

       6. sqrt-prod 22.3%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 28.8%

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

       9. sqrt-pow1 34.4%

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

       10. metadata-eval 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-*l* 35.0%

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

   15. Simplified 35.0%

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

   if 0.00121999999999999995 < k

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-/r* 32.2%

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

   3. Simplified 38.1%

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

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.0%

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

      3. times-frac 38.0%

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

   6. Applied egg-rr 89.1%

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

      1. associate-/r/ 89.1%

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

      2. associate-/r* 89.1%

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

      3. associate-/r/ 89.1%

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

   8. Simplified 89.1%

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

      1. frac-times 81.9%

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

      2. associate-/l* 81.9%

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

      3. div-inv 81.8%

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

      4. pow-flip 81.8%

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

      5. metadata-eval 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. times-frac 88.9%

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

       2. associate-*r/ 89.0%

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

       3. associate-/r* 88.9%

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

       4. associate-*l/ 81.8%

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

       5. unpow2 81.8%

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

   12. Simplified 94.5%

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

       1. unpow2 94.5%

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

       2. associate-/l/ 94.5%

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

       3. *-commutative 94.5%

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

       4. associate-/l/ 94.5%

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

       5. *-commutative 94.5%

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

   14. Applied egg-rr 94.5%

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

       1. unpow2 94.5%

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

   16. Simplified 94.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00122:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2}}{t}, \frac{2}{t}\right)} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\tan k \cdot \sin k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\tan k \cdot \sin k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0048:\\ \;\;\;\;{\left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k\_m}^{2}}{t\_m}, \frac{2}{t\_m}\right)} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left(t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}\right)} \cdot \left(l\_m \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k\_m}{t\_m}\right)}^{-2}}}{\sqrt[3]{\tan k\_m \cdot \sin k\_m} \cdot \left({\left(\sqrt[3]{l\_m}\right)}^{-2} \cdot t\_m\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.0048)
    (pow
     (*
      l_m
      (*
       (sqrt (fma -0.3333333333333333 (/ (pow k_m 2.0) t_m) (/ 2.0 t_m)))
       (pow k_m -2.0)))
     2.0)
    (if (<= k_m 2.35e+151)
      (*
       (/ 2.0 (* (pow k_m 2.0) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m)))))
       (* l_m l_m))
      (pow
       (/
        (cbrt (* 2.0 (pow (/ k_m t_m) -2.0)))
        (* (cbrt (* (tan k_m) (sin k_m))) (* (pow (cbrt l_m) -2.0) t_m)))
       3.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.0048) {
		tmp = pow((l_m * (sqrt(fma(-0.3333333333333333, (pow(k_m, 2.0) / t_m), (2.0 / t_m))) * pow(k_m, -2.0))), 2.0);
	} else if (k_m <= 2.35e+151) {
		tmp = (2.0 / (pow(k_m, 2.0) * (t_m * (pow(sin(k_m), 2.0) / cos(k_m))))) * (l_m * l_m);
	} else {
		tmp = pow((cbrt((2.0 * pow((k_m / t_m), -2.0))) / (cbrt((tan(k_m) * sin(k_m))) * (pow(cbrt(l_m), -2.0) * t_m))), 3.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.0048)
		tmp = Float64(l_m * Float64(sqrt(fma(-0.3333333333333333, Float64((k_m ^ 2.0) / t_m), Float64(2.0 / t_m))) * (k_m ^ -2.0))) ^ 2.0;
	elseif (k_m <= 2.35e+151)
		tmp = Float64(Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m))))) * Float64(l_m * l_m));
	else
		tmp = Float64(cbrt(Float64(2.0 * (Float64(k_m / t_m) ^ -2.0))) / Float64(cbrt(Float64(tan(k_m) * sin(k_m))) * Float64((cbrt(l_m) ^ -2.0) * t_m))) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.0048], N[Power[N[(l$95$m * N[(N[Sqrt[N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 2.35e+151], N[(N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(2.0 * N[Power[N[(k$95$m / t$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 0.00479999999999999958

   1. Initial program 35.4%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 27.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-/l* 26.6%

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

      4. associate-*r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. associate-/l* 26.6%

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

   7. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. associate-*r/ 27.7%

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

      4. pow2 27.7%

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

      5. pow-prod-down 47.5%

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

      6. pow2 47.5%

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

      7. *-commutative 47.5%

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

      8. pow2 47.5%

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

      9. pow-flip 47.5%

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

      10. metadata-eval 47.5%

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

   9. Applied egg-rr 47.5%

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

       1. *-commutative 47.5%

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

       2. fma-define 47.5%

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

       3. associate-*r/ 47.5%

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

       4. unpow2 47.5%

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

       5. swap-sqr 27.7%

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

       6. unpow2 27.7%

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

       7. unpow2 27.7%

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

       8. *-commutative 27.7%

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

       9. associate-*r* 27.7%

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

       10. associate-*l/ 26.6%

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

       11. associate-*r/ 26.6%

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

       12. distribute-rgt-in 43.8%

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

       13. metadata-eval 43.8%

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

       14. associate-*r/ 43.8%

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

       15. fma-define 43.8%

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

   11. Simplified 43.8%

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

       1. add-sqr-sqrt 24.4%

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

       2. pow2 24.4%

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

       3. *-commutative 24.4%

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

       4. pow2 24.4%

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

       5. sqrt-prod 24.4%

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

       6. sqrt-prod 22.3%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 28.8%

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

       9. sqrt-pow1 34.4%

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

       10. metadata-eval 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-*l* 35.0%

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

   15. Simplified 35.0%

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

   if 0.00479999999999999958 < k < 2.34999999999999995e151

   1. Initial program 25.6%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in t around 0 91.1%

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

      1. associate-/l* 91.0%

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

      2. associate-/l* 91.1%

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

   7. Simplified 91.1%

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

   if 2.34999999999999995e151 < k

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 40.5%

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

      1. add-cube-cbrt 40.4%

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

      2. pow3 40.4%

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

      3. cbrt-prod 40.5%

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

      4. cbrt-div 40.4%

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

      5. rem-cbrt-cube 57.2%

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

      6. cbrt-prod 67.4%

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

      7. pow2 67.4%

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

   6. Applied egg-rr 67.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 67.3%

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

   8. Applied egg-rr 80.9%

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

      1. unpow2 80.9%

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

      2. unpow3 80.9%

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

   10. Simplified 80.9%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0048:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k}^{2}}{t}, \frac{2}{t}\right)} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{\sqrt[3]{\tan k \cdot \sin k} \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00185:\\ \;\;\;\;{\left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-0.3333333333333333, \frac{{k\_m}^{2}}{t\_m}, \frac{2}{t\_m}\right)} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left(t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}\right)} \cdot \left(l\_m \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 0.00185)
    (pow
     (*
      l_m
      (*
       (sqrt (fma -0.3333333333333333 (/ (pow k_m 2.0) t_m) (/ 2.0 t_m)))
       (pow k_m -2.0)))
     2.0)
    (*
     (/ 2.0 (* (pow k_m 2.0) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m)))))
     (* l_m l_m)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 0.00185) {
		tmp = pow((l_m * (sqrt(fma(-0.3333333333333333, (pow(k_m, 2.0) / t_m), (2.0 / t_m))) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = (2.0 / (pow(k_m, 2.0) * (t_m * (pow(sin(k_m), 2.0) / cos(k_m))))) * (l_m * l_m);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 0.00185)
		tmp = Float64(l_m * Float64(sqrt(fma(-0.3333333333333333, Float64((k_m ^ 2.0) / t_m), Float64(2.0 / t_m))) * (k_m ^ -2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m))))) * Float64(l_m * l_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 0.00185], N[Power[N[(l$95$m * N[(N[Sqrt[N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.0018500000000000001

   1. Initial program 35.4%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in k around 0 27.7%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-/l* 26.6%

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

      4. associate-*r/ 26.6%

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

      5. *-commutative 26.6%

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

      6. associate-/l* 26.6%

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

   7. Simplified 26.6%

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

      1. div-inv 26.6%

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

      2. pow2 26.6%

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

      3. associate-*r/ 27.7%

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

      4. pow2 27.7%

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

      5. pow-prod-down 47.5%

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

      6. pow2 47.5%

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

      7. *-commutative 47.5%

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

      8. pow2 47.5%

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

      9. pow-flip 47.5%

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

      10. metadata-eval 47.5%

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

   9. Applied egg-rr 47.5%

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

       1. *-commutative 47.5%

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

       2. fma-define 47.5%

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

       3. associate-*r/ 47.5%

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

       4. unpow2 47.5%

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

       5. swap-sqr 27.7%

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

       6. unpow2 27.7%

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

       7. unpow2 27.7%

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

       8. *-commutative 27.7%

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

       9. associate-*r* 27.7%

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

       10. associate-*l/ 26.6%

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

       11. associate-*r/ 26.6%

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

       12. distribute-rgt-in 43.8%

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

       13. metadata-eval 43.8%

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

       14. associate-*r/ 43.8%

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

       15. fma-define 43.8%

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

   11. Simplified 43.8%

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

       1. add-sqr-sqrt 24.4%

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

       2. pow2 24.4%

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

       3. *-commutative 24.4%

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

       4. pow2 24.4%

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

       5. sqrt-prod 24.4%

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

       6. sqrt-prod 22.3%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 28.8%

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

       9. sqrt-pow1 34.4%

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

       10. metadata-eval 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-*l* 35.0%

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

   15. Simplified 35.0%

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

   if 0.0018500000000000001 < k

   1. Initial program 32.2%

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

   3. Simplified 38.2%

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

   5. Taylor expanded in t around 0 77.2%

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

      1. associate-/l* 77.2%

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

      2. associate-/l* 77.2%

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

   7. Simplified 77.2%

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

Alternative 8: 74.0% accurate, 1.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (*
   (/ 2.0 (* (pow k_m 2.0) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m)))))
   (* l_m l_m))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((2.0 / (pow(k_m, 2.0) * (t_m * (pow(sin(k_m), 2.0) / cos(k_m))))) * (l_m * l_m));
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((2.0d0 / ((k_m ** 2.0d0) * (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m))))) * (l_m * l_m))
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((2.0 / (Math.pow(k_m, 2.0) * (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))))) * (l_m * l_m));
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((2.0 / (math.pow(k_m, 2.0) * (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))) * (l_m * l_m))

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m))))) * Float64(l_m * l_m)))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((2.0 / ((k_m ^ 2.0) * (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))))) * (l_m * l_m));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * N[(N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Simplified 40.5%

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

5. Taylor expanded in t around 0 74.9%

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

   1. associate-/l* 74.9%

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

   2. associate-/l* 74.9%

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

7. Simplified 74.9%

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

Alternative 9: 74.0% accurate, 1.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (*
   (* l_m l_m)
   (* 2.0 (/ (cos k_m) (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m * l_m) * (2.0 * (cos(k_m) / (pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))))));
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m * l_m) * (2.0d0 * (cos(k_m) / ((k_m ** 2.0d0) * (t_m * (sin(k_m) ** 2.0d0))))))
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m * l_m) * (2.0 * (Math.cos(k_m) / (Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))))));
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m * l_m) * (2.0 * (math.cos(k_m) / (math.pow(k_m, 2.0) * (t_m * math.pow(math.sin(k_m), 2.0))))))

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m * l_m) * Float64(2.0 * Float64(cos(k_m) / Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0)))))))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m * l_m) * (2.0 * (cos(k_m) / ((k_m ^ 2.0) * (t_m * (sin(k_m) ^ 2.0))))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Simplified 40.5%

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

5. Taylor expanded in t around 0 74.9%

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

6. Final simplification 74.9%

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

Alternative 10: 65.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(2 \cdot \frac{1}{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}\right)\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (*
   (* l_m l_m)
   (* 2.0 (/ 1.0 (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m * l_m) * (2.0 * (1.0 / (pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))))));
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m * l_m) * (2.0d0 * (1.0d0 / ((k_m ** 2.0d0) * (t_m * (sin(k_m) ** 2.0d0))))))
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m * l_m) * (2.0 * (1.0 / (Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))))));
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m * l_m) * (2.0 * (1.0 / (math.pow(k_m, 2.0) * (t_m * math.pow(math.sin(k_m), 2.0))))))

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m * l_m) * Float64(2.0 * Float64(1.0 / Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0)))))))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m * l_m) * (2.0 * (1.0 / ((k_m ^ 2.0) * (t_m * (sin(k_m) ^ 2.0))))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 * N[(1.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Simplified 40.5%

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

5. Taylor expanded in t around 0 74.9%

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

6. Taylor expanded in k around 0 66.4%

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

7. Final simplification 66.4%

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

Alternative 11: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot \frac{{l\_m}^{2} \cdot {k\_m}^{-4}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{l\_m}^{2}}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.6e+51)
    (* 2.0 (/ (* (pow l_m 2.0) (pow k_m -4.0)) t_m))
    (* -0.3333333333333333 (/ (pow l_m 2.0) (* t_m (pow k_m 2.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.6e+51) {
		tmp = 2.0 * ((pow(l_m, 2.0) * pow(k_m, -4.0)) / t_m);
	} else {
		tmp = -0.3333333333333333 * (pow(l_m, 2.0) / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.6d+51) then
        tmp = 2.0d0 * (((l_m ** 2.0d0) * (k_m ** (-4.0d0))) / t_m)
    else
        tmp = (-0.3333333333333333d0) * ((l_m ** 2.0d0) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.6e+51) {
		tmp = 2.0 * ((Math.pow(l_m, 2.0) * Math.pow(k_m, -4.0)) / t_m);
	} else {
		tmp = -0.3333333333333333 * (Math.pow(l_m, 2.0) / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 4.6e+51:
		tmp = 2.0 * ((math.pow(l_m, 2.0) * math.pow(k_m, -4.0)) / t_m)
	else:
		tmp = -0.3333333333333333 * (math.pow(l_m, 2.0) / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 4.6e+51)
		tmp = Float64(2.0 * Float64(Float64((l_m ^ 2.0) * (k_m ^ -4.0)) / t_m));
	else
		tmp = Float64(-0.3333333333333333 * Float64((l_m ^ 2.0) / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 4.6e+51)
		tmp = 2.0 * (((l_m ^ 2.0) * (k_m ^ -4.0)) / t_m);
	else
		tmp = -0.3333333333333333 * ((l_m ^ 2.0) / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.6e+51], N[(2.0 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.6000000000000001e51

   1. Initial program 33.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/r* 63.4%

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

   7. Simplified 63.4%

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

      1. div-inv 63.4%

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

      2. pow-flip 63.4%

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

      3. metadata-eval 63.4%

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

   9. Applied egg-rr 63.4%

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

       1. pow1 63.4%

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

       2. associate-*l* 63.5%

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

       3. pow2 63.5%

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

   11. Applied egg-rr 63.5%

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

       1. unpow1 63.5%

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

       2. associate-*l/ 63.4%

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

       3. associate-/l* 63.4%

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

       4. *-commutative 63.4%

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

   13. Simplified 63.4%

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

   if 4.6000000000000001e51 < k

   1. Initial program 37.9%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in k around 0 15.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 15.0%

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

      2. *-commutative 15.0%

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

      3. associate-/l* 10.9%

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

      4. associate-*r/ 10.9%

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

      5. *-commutative 10.9%

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

      6. associate-/l* 10.9%

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

   7. Simplified 10.9%

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

   8. Taylor expanded in k around inf 62.3%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot {k}^{-4}}{t}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{l\_m}^{2}}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.6e+51)
    (* (* l_m l_m) (* (/ 2.0 t_m) (pow k_m -4.0)))
    (* -0.3333333333333333 (/ (pow l_m 2.0) (* t_m (pow k_m 2.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.6e+51) {
		tmp = (l_m * l_m) * ((2.0 / t_m) * pow(k_m, -4.0));
	} else {
		tmp = -0.3333333333333333 * (pow(l_m, 2.0) / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.6d+51) then
        tmp = (l_m * l_m) * ((2.0d0 / t_m) * (k_m ** (-4.0d0)))
    else
        tmp = (-0.3333333333333333d0) * ((l_m ** 2.0d0) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.6e+51) {
		tmp = (l_m * l_m) * ((2.0 / t_m) * Math.pow(k_m, -4.0));
	} else {
		tmp = -0.3333333333333333 * (Math.pow(l_m, 2.0) / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 4.6e+51:
		tmp = (l_m * l_m) * ((2.0 / t_m) * math.pow(k_m, -4.0))
	else:
		tmp = -0.3333333333333333 * (math.pow(l_m, 2.0) / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 4.6e+51)
		tmp = Float64(Float64(l_m * l_m) * Float64(Float64(2.0 / t_m) * (k_m ^ -4.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64((l_m ^ 2.0) / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 4.6e+51)
		tmp = (l_m * l_m) * ((2.0 / t_m) * (k_m ^ -4.0));
	else
		tmp = -0.3333333333333333 * ((l_m ^ 2.0) / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.6e+51], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.6000000000000001e51

   1. Initial program 33.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/r* 63.4%

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

   7. Simplified 63.4%

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

      1. div-inv 63.4%

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

      2. pow-flip 63.4%

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

      3. metadata-eval 63.4%

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

   9. Applied egg-rr 63.4%

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

   if 4.6000000000000001e51 < k

   1. Initial program 37.9%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in k around 0 15.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 15.0%

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

      2. *-commutative 15.0%

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

      3. associate-/l* 10.9%

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

      4. associate-*r/ 10.9%

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

      5. *-commutative 10.9%

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

      6. associate-/l* 10.9%

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

   7. Simplified 10.9%

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

   8. Taylor expanded in k around inf 62.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{l\_m}^{2}}{{k\_m}^{2}}}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.6e+51)
    (* (* l_m l_m) (* (/ 2.0 t_m) (pow k_m -4.0)))
    (* -0.3333333333333333 (/ (/ (pow l_m 2.0) (pow k_m 2.0)) t_m)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.6e+51) {
		tmp = (l_m * l_m) * ((2.0 / t_m) * pow(k_m, -4.0));
	} else {
		tmp = -0.3333333333333333 * ((pow(l_m, 2.0) / pow(k_m, 2.0)) / t_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.6d+51) then
        tmp = (l_m * l_m) * ((2.0d0 / t_m) * (k_m ** (-4.0d0)))
    else
        tmp = (-0.3333333333333333d0) * (((l_m ** 2.0d0) / (k_m ** 2.0d0)) / t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.6e+51) {
		tmp = (l_m * l_m) * ((2.0 / t_m) * Math.pow(k_m, -4.0));
	} else {
		tmp = -0.3333333333333333 * ((Math.pow(l_m, 2.0) / Math.pow(k_m, 2.0)) / t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 4.6e+51:
		tmp = (l_m * l_m) * ((2.0 / t_m) * math.pow(k_m, -4.0))
	else:
		tmp = -0.3333333333333333 * ((math.pow(l_m, 2.0) / math.pow(k_m, 2.0)) / t_m)
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 4.6e+51)
		tmp = Float64(Float64(l_m * l_m) * Float64(Float64(2.0 / t_m) * (k_m ^ -4.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64((l_m ^ 2.0) / (k_m ^ 2.0)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 4.6e+51)
		tmp = (l_m * l_m) * ((2.0 / t_m) * (k_m ^ -4.0));
	else
		tmp = -0.3333333333333333 * (((l_m ^ 2.0) / (k_m ^ 2.0)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.6e+51], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.6000000000000001e51

   1. Initial program 33.6%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-/r* 63.4%

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

   7. Simplified 63.4%

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

      1. div-inv 63.4%

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

      2. pow-flip 63.4%

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

      3. metadata-eval 63.4%

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

   9. Applied egg-rr 63.4%

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

   if 4.6000000000000001e51 < k

   1. Initial program 37.9%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in k around 0 15.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. fma-define 15.0%

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

      2. *-commutative 15.0%

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

      3. associate-/l* 10.9%

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

      4. associate-*r/ 10.9%

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

      5. *-commutative 10.9%

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

      6. associate-/l* 10.9%

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

   7. Simplified 10.9%

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

      1. div-inv 10.9%

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

      2. pow2 10.9%

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

      3. associate-*r/ 15.0%

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

      4. pow2 15.0%

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

      5. pow-prod-down 40.0%

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

      6. pow2 40.0%

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

      7. *-commutative 40.0%

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

      8. pow2 40.0%

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

      9. pow-flip 40.0%

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

      10. metadata-eval 40.0%

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

   9. Applied egg-rr 40.0%

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

       1. *-commutative 40.0%

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

       2. fma-define 40.0%

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

       3. associate-*r/ 40.0%

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

       4. unpow2 40.0%

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

       5. swap-sqr 15.0%

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

       6. unpow2 15.0%

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

       7. unpow2 15.0%

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

       8. *-commutative 15.0%

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

       9. associate-*r* 15.0%

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

       10. associate-*l/ 10.9%

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

       11. associate-*r/ 10.9%

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

       12. distribute-rgt-in 12.7%

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

       13. metadata-eval 12.7%

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

       14. associate-*r/ 12.7%

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

       15. fma-define 12.7%

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

   11. Simplified 12.7%

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

   12. Taylor expanded in k around inf 62.3%

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

       1. associate-/r* 62.4%

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

   14. Simplified 62.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+51}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t} \cdot {k}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \left(\frac{2}{t\_m} \cdot {k\_m}^{-4}\right)\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m)
 :precision binary64
 (* t_s (* (* l_m l_m) (* (/ 2.0 t_m) (pow k_m -4.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m * l_m) * ((2.0 / t_m) * pow(k_m, -4.0)));
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * ((l_m * l_m) * ((2.0d0 / t_m) * (k_m ** (-4.0d0))))
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * ((l_m * l_m) * ((2.0 / t_m) * Math.pow(k_m, -4.0)));
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * ((l_m * l_m) * ((2.0 / t_m) * math.pow(k_m, -4.0)))

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m * l_m) * Float64(Float64(2.0 / t_m) * (k_m ^ -4.0))))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m * l_m) * ((2.0 / t_m) * (k_m ^ -4.0)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Simplified 40.5%

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

5. Taylor expanded in k around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-/r* 62.2%

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

7. Simplified 62.2%

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

   1. div-inv 62.2%

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

   2. pow-flip 62.2%

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

   3. metadata-eval 62.2%

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

9. Applied egg-rr 62.2%

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

10. Final simplification 62.2%

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

Alternative 15: 29.0% accurate, 421.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 0 \end{array} \]

FPCore

l_m = (fabs.f64 l)
k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k_m) :precision binary64 (* t_s 0.0))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * 0.0;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * 0.0d0
end function

Java

l_m = Math.abs(l);
k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * 0.0;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * 0.0

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * 0.0)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * 0.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l$95$m_, k$95$m_] := N[(t$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 34.5%

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

3. Simplified 40.5%

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

5. Taylor expanded in k around 0 62.2%

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

   1. *-commutative 62.2%

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

   2. associate-/r* 62.2%

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

7. Simplified 62.2%

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

   1. add-log-exp 58.4%

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

   2. div-inv 58.4%

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

   3. exp-prod 46.8%

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

   4. pow-flip 46.8%

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

   5. metadata-eval 46.8%

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

9. Applied egg-rr 46.8%

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

10. Taylor expanded in t around inf 33.3%

    \[\leadsto \log \color{blue}{1} \cdot \left(\ell \cdot \ell\right) \]

11. Taylor expanded in l around 0 33.9%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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))))