Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 84.6%

Time: 18.7s

Alternatives: 15

Speedup: 12.5×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(t\_m, t\_m \cdot 2, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e-90)
    (/
     2.0
     (*
      t_m
      (*
       (/ (pow (sin k) 2.0) (* l (* l (cos k))))
       (fma t_m (* t_m 2.0) (* k k)))))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e-90) {
		tmp = 2.0 / (t_m * ((pow(sin(k), 2.0) / (l * (l * cos(k)))) * fma(t_m, (t_m * 2.0), (k * k))));
	} else {
		tmp = (2.0 / (t_m / l)) / (((t_m / l) * (t_m * sin(k))) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7e-90)
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64((sin(k) ^ 2.0) / Float64(l * Float64(l * cos(k)))) * fma(t_m, Float64(t_m * 2.0), Float64(k * k)))));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7e-90], N[(2.0 / N[(t$95$m * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(t$95$m * 2.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.9999999999999997e-90

   1. Initial program 44.1%

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

   3. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

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

   5. Simplified 63.5%

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

   if 6.9999999999999997e-90 < t

   1. Initial program 72.2%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 83.9

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

   4. Applied egg-rr 83.9%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 83.9

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

   6. Applied egg-rr 83.9%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 91.9%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 91.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(t, t \cdot 2, k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (*
        (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))
        (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))
       INFINITY)
    (* l (/ l (* (* t_m k) (* k (* t_m t_m)))))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (((tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))) * (1.0 + (1.0 + pow((k / t_m), 2.0)))) <= ((double) INFINITY)) {
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (((Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0)))) <= Double.POSITIVE_INFINITY) {
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if ((math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))) * (1.0 + (1.0 + math.pow((k / t_m), 2.0)))) <= math.inf:
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))) <= Inf)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(k * Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (((tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))) * (1.0 + (1.0 + ((k / t_m) ^ 2.0)))) <= Inf)
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.8%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 70.0

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

   5. Simplified 70.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 72.6

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

   7. Applied egg-rr 72.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 80.1

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

   9. Applied egg-rr 80.1%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 12.9

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

   5. Simplified 12.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 27.3

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

   7. Applied egg-rr 27.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 27.6

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

   9. Applied egg-rr 27.6%

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

   10. Taylor expanded in l around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 42.6

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

   12. Simplified 42.6%

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

4. Final simplification 67.4%

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

Alternative 3: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\sin k \cdot \frac{t\_m \cdot \left(k \cdot \left(k \cdot \sin k\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-90)
    (/ 2.0 (* (sin k) (/ (* t_m (* k (* k (sin k)))) (* l (* l (cos k))))))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5e-90) {
		tmp = 2.0 / (sin(k) * ((t_m * (k * (k * sin(k)))) / (l * (l * cos(k)))));
	} else {
		tmp = (2.0 / (t_m / l)) / (((t_m / l) * (t_m * sin(k))) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5e-90)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(t_m * Float64(k * Float64(k * sin(k)))) / Float64(l * Float64(l * cos(k))))));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-90], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(t$95$m * N[(k * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000019e-90

   1. Initial program 44.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 33.0%

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

   5. Taylor expanded in k around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 59.8

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

   7. Simplified 59.8%

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

   if 5.00000000000000019e-90 < t

   1. Initial program 72.2%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 83.9

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

   4. Applied egg-rr 83.9%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 83.9

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

   6. Applied egg-rr 83.9%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 91.9%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 91.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{\sin k \cdot \frac{t \cdot \left(k \cdot \left(k \cdot \sin k\right)\right)}{\ell \cdot \left(\ell \cdot \cos k\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{t\_m \cdot \left(k \cdot \left(k \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-147)
    (/ (* (cos k) (* 2.0 (* l l))) (* t_m (* k (* k (pow (sin k) 2.0)))))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.2e-147) {
		tmp = (cos(k) * (2.0 * (l * l))) / (t_m * (k * (k * pow(sin(k), 2.0))));
	} else {
		tmp = (2.0 / (t_m / l)) / (((t_m / l) * (t_m * sin(k))) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.2e-147)
		tmp = Float64(Float64(cos(k) * Float64(2.0 * Float64(l * l))) / Float64(t_m * Float64(k * Float64(k * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.2e-147], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(k * N[(k * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.19999999999999979e-147

   1. Initial program 45.5%

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

      1. associate-*l/ N/A

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

      2. cube-mult N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 58.0

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

   4. Applied egg-rr 58.0%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. pow-lowering-pow.f64 N/A

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

      19. sin-lowering-sin.f64 60.5

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

   7. Simplified 60.5%

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

   if 3.19999999999999979e-147 < t

   1. Initial program 66.0%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 81.0

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

   4. Applied egg-rr 81.0%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 81.0

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

   6. Applied egg-rr 81.0%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 87.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\cos k \cdot \left(2 \cdot \left(\ell \cdot \ell\right)\right)}{t \cdot \left(k \cdot \left(k \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-147)
    (* (* 2.0 (* l l)) (/ (cos k) (* (pow (sin k) 2.0) (* t_m (* k k)))))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.95e-147) {
		tmp = (2.0 * (l * l)) * (cos(k) / (pow(sin(k), 2.0) * (t_m * (k * k))));
	} else {
		tmp = (2.0 / (t_m / l)) / (((t_m / l) * (t_m * sin(k))) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.95e-147)
		tmp = Float64(Float64(2.0 * Float64(l * l)) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * Float64(t_m * Float64(k * k)))));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-147], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9499999999999999e-147

   1. Initial program 45.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 33.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow-lowering-pow.f64 N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 61.8

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

   7. Simplified 61.8%

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

   if 1.9499999999999999e-147 < t

   1. Initial program 66.0%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 81.0

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

   4. Applied egg-rr 81.0%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 81.0

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

   6. Applied egg-rr 81.0%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 87.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right) \cdot \frac{t\_m \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.15e-147)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (if (<= t_m 1.4e+105)
      (/
       2.0
       (*
        (sin k)
        (*
         (* (/ t_m l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))
         (/ (* t_m t_m) l))))
      (/
       2.0
       (* 2.0 (* (/ t_m l) (* (tan k) (* t_m (/ (* t_m (sin k)) l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.15e-147) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else if (t_m <= 1.4e+105) {
		tmp = 2.0 / (sin(k) * (((t_m / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 / (2.0 * ((t_m / l) * (tan(k) * (t_m * ((t_m * sin(k)) / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.15e-147)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	elseif (t_m <= 1.4e+105)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(Float64(t_m / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) * Float64(Float64(t_m * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t_m / l) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.15e-147], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+105], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.14999999999999995e-147

   1. Initial program 45.5%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.4

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

   5. Simplified 47.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 54.6

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

   7. Applied egg-rr 54.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 58.4

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

   9. Applied egg-rr 58.4%

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

   10. Taylor expanded in l around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 61.2

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

   12. Simplified 61.2%

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

   if 1.14999999999999995e-147 < t < 1.4000000000000001e105

   1. Initial program 63.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 63.5%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 83.5%

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

   if 1.4000000000000001e105 < t

   1. Initial program 69.8%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 80.1

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

   4. Applied egg-rr 80.1%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 80.1

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

   6. Applied egg-rr 80.1%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 97.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right) \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-147)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-147) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = (2.0 / (t_m / l)) / (((t_m / l) * (t_m * sin(k))) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-147)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-147], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.1000000000000001e-147

   1. Initial program 45.5%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.4

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

   5. Simplified 47.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 54.6

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

   7. Applied egg-rr 54.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 58.4

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

   9. Applied egg-rr 58.4%

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

   10. Taylor expanded in l around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 61.2

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

   12. Simplified 61.2%

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

   if 1.1000000000000001e-147 < t

   1. Initial program 66.0%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 81.0

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

   4. Applied egg-rr 81.0%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 81.0

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

   6. Applied egg-rr 81.0%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 87.8%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 88.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{\ell}}}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-147)
    (* l (/ l (* t_m (* t_m (* t_m (* k k))))))
    (/
     2.0
     (*
      (* (/ t_m l) (fma k (/ k (* t_m t_m)) 2.0))
      (* (tan k) (* t_m (/ (* t_m (sin k)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.95e-147) {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	} else {
		tmp = 2.0 / (((t_m / l) * fma(k, (k / (t_m * t_m)), 2.0)) * (tan(k) * (t_m * ((t_m * sin(k)) / l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.95e-147)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-147], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9499999999999999e-147

   1. Initial program 45.5%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.4

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

   5. Simplified 47.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 54.6

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

   7. Applied egg-rr 54.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 58.4

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

   9. Applied egg-rr 58.4%

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

   10. Taylor expanded in l around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 N/A

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

       13. *-commutative N/A

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

       14. *-lowering-*.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 61.2

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

   12. Simplified 61.2%

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

   if 1.9499999999999999e-147 < t

   1. Initial program 66.0%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 81.0

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

   4. Applied egg-rr 81.0%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 81.0

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

   6. Applied egg-rr 81.0%

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

   8. Applied egg-rr 87.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 2.7e+71)
    (/
     2.0
     (*
      (* (/ t_m l) (* (tan k) (* t_m (/ (* t_m k) l))))
      (fma (/ k t_m) (/ k t_m) 2.0)))
    (/ 2.0 (* 2.0 (* (/ t_m l) (* (tan k) (* t_m (/ (* t_m (sin k)) l)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 2.7e+71) {
		tmp = 2.0 / (((t_m / l) * (tan(k) * (t_m * ((t_m * k) / l)))) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = 2.0 / (2.0 * ((t_m / l) * (tan(k) * (t_m * ((t_m * sin(k)) / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 2.7e+71)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * k) / l)))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(t_m / l) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 2.7e+71], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.69999999999999997e71

   1. Initial program 57.1%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 67.3

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

   4. Applied egg-rr 67.3%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 67.3

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

   6. Applied egg-rr 67.3%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 79.1%

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

   9. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 77.1

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

   11. Simplified 77.1%

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

   if 2.69999999999999997e71 < l

   1. Initial program 38.0%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 53.8

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

   4. Applied egg-rr 53.8%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 53.8

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

   6. Applied egg-rr 53.8%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 61.1%

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

   9. Taylor expanded in k around 0

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

   11. Simplified 68.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 1.6e-42)
    (/
     2.0
     (*
      (* (/ t_m l) (* (tan k) (* t_m (/ (* t_m k) l))))
      (fma (/ k t_m) (/ k t_m) 2.0)))
    (* (/ l (* t_m (* k (* t_m k)))) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 1.6e-42) {
		tmp = 2.0 / (((t_m / l) * (tan(k) * (t_m * ((t_m * k) / l)))) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = (l / (t_m * (k * (t_m * k)))) * (l / t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (l <= 1.6e-42)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * k) / l)))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(Float64(l / Float64(t_m * Float64(k * Float64(t_m * k)))) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[l, 1.6e-42], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.60000000000000012e-42

   1. Initial program 53.7%

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

      1. unpow3 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 65.6

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

   4. Applied egg-rr 65.6%

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 65.6

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

   6. Applied egg-rr 65.6%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. frac-times N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   8. Applied egg-rr 78.4%

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

   9. Taylor expanded in k around 0

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

       1. /-lowering-/.f64 N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 76.1

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

   11. Simplified 76.1%

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

   if 1.60000000000000012e-42 < l

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 55.9

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

   5. Simplified 55.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 61.4

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

   7. Applied egg-rr 61.4%

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 67.5

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

   9. Applied egg-rr 67.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{t\_m}}{t\_m \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.5e-144)
    (* (/ l (* t_m (* t_m k))) (/ l (* t_m k)))
    (* (/ l t_m) (/ (/ l t_m) (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e-144) {
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k));
	} else {
		tmp = (l / t_m) * ((l / t_m) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.5d-144) then
        tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k))
    else
        tmp = (l / t_m) * ((l / t_m) / (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.5e-144) {
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k));
	} else {
		tmp = (l / t_m) * ((l / t_m) / (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.5e-144:
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k))
	else:
		tmp = (l / t_m) * ((l / t_m) / (t_m * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.5e-144)
		tmp = Float64(Float64(l / Float64(t_m * Float64(t_m * k))) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / t_m) / Float64(t_m * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.5e-144)
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k));
	else
		tmp = (l / t_m) * ((l / t_m) / (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 1.5e-144], N[(N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4999999999999999e-144

   1. Initial program 56.3%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 49.8

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

   5. Simplified 49.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 56.6

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

   7. Applied egg-rr 56.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 65.4

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

   9. Applied egg-rr 65.4%

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

       1. associate-*l/ N/A

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

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k}} \cdot \frac{\ell}{k \cdot t} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k} \cdot \frac{\ell}{k \cdot t}} \]

       6. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot t}} \cdot \frac{\ell}{k \cdot t} \]

       9. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \frac{\ell}{k \cdot t} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \frac{\ell}{k \cdot t} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       14. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

       15. *-lowering-*.f64 72.7

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

   11. Applied egg-rr 72.7%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}} \]

   if 1.4999999999999999e-144 < k

   1. Initial program 48.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 51.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 58.2

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 58.2%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{t \cdot \left(k \cdot k\right)}} \cdot \ell \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t} \cdot \ell}{t \cdot \left(k \cdot k\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k \cdot k} \cdot \frac{\ell}{t}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k \cdot k} \cdot \frac{\ell}{t}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k \cdot k}} \cdot \frac{\ell}{t} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\ell}{t \cdot t}}}{k \cdot k} \cdot \frac{\ell}{t} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{\color{blue}{t \cdot t}}}{k \cdot k} \cdot \frac{\ell}{t} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\ell}{t \cdot t}}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{t} \]

      10. /-lowering-/.f64 59.1

          \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k \cdot k} \cdot \color{blue}{\frac{\ell}{t}} \]

   9. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k \cdot k} \cdot \frac{\ell}{t}} \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t}}{t}}}{k \cdot k} \cdot \frac{\ell}{t} \]

       2. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot t}} \cdot \frac{\ell}{t} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{t} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{\left(k \cdot t\right) \cdot k}} \cdot \frac{\ell}{t} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(k \cdot t\right) \cdot k}} \cdot \frac{\ell}{t} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\ell}{t}}}{\left(k \cdot t\right) \cdot k} \cdot \frac{\ell}{t} \]

       7. *-commutative N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{\left(t \cdot k\right)} \cdot k} \cdot \frac{\ell}{t} \]

       8. associate-*l* N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{t} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{t} \]

       10. *-lowering-*.f64 66.4

           \[\leadsto \frac{\frac{\ell}{t}}{t \cdot \color{blue}{\left(k \cdot k\right)}} \cdot \frac{\ell}{t} \]

   11. Applied egg-rr 66.4%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}} \cdot \frac{\ell}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2e+106)
    (* (/ l (* t_m (* t_m k))) (/ l (* t_m k)))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e+106) {
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2d+106) then
        tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k))
    else
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e+106) {
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2e+106:
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e+106)
		tmp = Float64(Float64(l / Float64(t_m * Float64(t_m * k))) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2e+106)
		tmp = (l / (t_m * (t_m * k))) * (l / (t_m * k));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 2e+106], N[(N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000018e106

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 52.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 52.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 59.1

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 59.1%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 65.7

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 65.7%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \]

       3. times-frac N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{k \cdot t}} \]

       4. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k}} \cdot \frac{\ell}{k \cdot t} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{t \cdot t}}{k} \cdot \frac{\ell}{k \cdot t}} \]

       6. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       8. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot t}} \cdot \frac{\ell}{k \cdot t} \]

       9. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{k \cdot t} \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \frac{\ell}{k \cdot t} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \frac{\ell}{k \cdot t} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

       14. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

       15. *-lowering-*.f64 72.4

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t \cdot k}} \]

   11. Applied egg-rr 72.4%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}} \]

   if 2.00000000000000018e106 < k

   1. Initial program 40.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 38.6

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 47.7

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 44.9

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 44.9%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \cdot \ell \]

       3. cube-mult N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \cdot \ell \]

       4. unpow2 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \cdot \ell \]

       5. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left({t}^{2} \cdot {k}^{2}\right)}} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left({k}^{2} \cdot {t}^{2}\right)}} \cdot \ell \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left({k}^{2} \cdot {t}^{2}\right)}} \cdot \ell \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}} \cdot \ell \]

       9. unpow2 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)} \cdot \ell \]

       10. associate-*l* N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \cdot \ell \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \cdot \ell \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \cdot \ell \]

       15. unpow2 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]

       16. *-lowering-*.f64 57.7

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]

   12. Simplified 57.7%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.8e-164)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.8e-164) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.8d-164) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.8e-164) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.8e-164:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.8e-164)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.8e-164)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 3.8e-164], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.79999999999999989e-164

   1. Initial program 56.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 50.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 55.9

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 65.0

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 65.0%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       9. *-lowering-*.f64 69.5

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   11. Applied egg-rr 69.5%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 3.79999999999999989e-164 < k

   1. Initial program 48.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 51.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{t}} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \]

      9. *-lowering-*.f64 65.4

         \[\leadsto \frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   7. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-95}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-95)
    (* l (/ l (* (* t_m k) (* t_m (* t_m k)))))
    (* l (/ l (* t_m (* t_m (* t_m (* k k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-95) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6d-95) then
        tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
    else
        tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-95) {
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	} else {
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6e-95:
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))))
	else:
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-95)
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(t_m * Float64(t_m * k)))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6e-95)
		tmp = l * (l / ((t_m * k) * (t_m * (t_m * k))));
	else
		tmp = l * (l / (t_m * (t_m * (t_m * (k * k)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_, k_] := N[(t$95$s * If[LessEqual[k, 6e-95], N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6e-95

   1. Initial program 57.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 53.0

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 59.3

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 67.5

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 67.5%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

       3. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot t\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       4. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{\left(t \cdot k\right)}\right) \cdot \left(k \cdot t\right)} \cdot \ell \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

       9. *-lowering-*.f64 71.5

          \[\leadsto \frac{\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \cdot \ell \]

   11. Applied egg-rr 71.5%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \left(t \cdot k\right)}} \cdot \ell \]

   if 6e-95 < k

   1. Initial program 45.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. *-lowering-*.f64 45.4

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Simplified 45.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

      5. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

      6. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

      10. *-lowering-*.f64 52.8

          \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

   7. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

      2. associate-*r* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

      3. associate-*l* N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

      8. *-lowering-*.f64 51.4

         \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

   9. Applied egg-rr 51.4%

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   10. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
   11. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]

       2. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \cdot \ell \]

       3. cube-mult N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \cdot \ell \]

       4. unpow2 N/A

          \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \cdot \ell \]

       5. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left({t}^{2} \cdot {k}^{2}\right)}} \cdot \ell \]

       6. *-commutative N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left({k}^{2} \cdot {t}^{2}\right)}} \cdot \ell \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left({k}^{2} \cdot {t}^{2}\right)}} \cdot \ell \]

       8. *-commutative N/A

          \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}} \cdot \ell \]

       9. unpow2 N/A

          \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)} \cdot \ell \]

       10. associate-*l* N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}} \cdot \ell \]

       11. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \cdot \ell \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \ell \]

       13. *-commutative N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \cdot \ell \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \cdot \ell \]

       15. unpow2 N/A

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]

       16. *-lowering-*.f64 59.0

           \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]

   12. Simplified 59.0%

       \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-95}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (t_m * (t_m * (k * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * (t_m * (t_m * (k * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * (t_m * (t_m * (k * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * (t_m * (t_m * (k * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * (t_m * (t_m * (k * k))))));
end

Wolfram

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_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. *-lowering-*.f64 50.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Simplified 50.6%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)} \cdot \ell} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   5. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \cdot \ell \]

   6. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot t\right)} \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}} \cdot \ell \]

   10. *-lowering-*.f64 57.2

       \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)} \cdot \ell \]

7. Applied egg-rr 57.2%

   \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

   2. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \left(t \cdot t\right)} \cdot \ell \]

   3. associate-*l* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   5. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right)} \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \color{blue}{\left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

   8. *-lowering-*.f64 62.3

      \[\leadsto \frac{\ell}{\left(k \cdot t\right) \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \ell \]

9. Applied egg-rr 62.3%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

10. Taylor expanded in l around 0

    \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\ell}{{k}^{2} \cdot {t}^{3}}} \cdot \ell \]

    2. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \cdot \ell \]

    3. cube-mult N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \cdot \ell \]

    4. unpow2 N/A

       \[\leadsto \frac{\ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \cdot \ell \]

    5. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left({t}^{2} \cdot {k}^{2}\right)}} \cdot \ell \]

    6. *-commutative N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left({k}^{2} \cdot {t}^{2}\right)}} \cdot \ell \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{t \cdot \left({k}^{2} \cdot {t}^{2}\right)}} \cdot \ell \]

    8. *-commutative N/A

       \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}} \cdot \ell \]

    9. unpow2 N/A

       \[\leadsto \frac{\ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)} \cdot \ell \]

    10. associate-*l* N/A

        \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}} \cdot \ell \]

    11. *-commutative N/A

        \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \cdot \ell \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \frac{\ell}{t \cdot \color{blue}{\left(t \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \ell \]

    13. *-commutative N/A

        \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \cdot \ell \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \cdot \ell \]

    15. unpow2 N/A

        \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]

    16. *-lowering-*.f64 62.5

        \[\leadsto \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]

12. Simplified 62.5%

    \[\leadsto \color{blue}{\frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \cdot \ell \]

13. Final simplification 62.5%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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))))