Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.3% → 92.7%

Time: 19.0s

Alternatives: 19

Speedup: 12.5×

• 27.5% 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: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(t\_2 \cdot \left(k \cdot \tan k\right), t\_m \cdot k, \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot 2\right)\right)\right) \cdot \left(t\_2 \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \left(\left(\left(t\_m \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t\_m \cdot \sin k\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 2.3e+44)
      (/
       2.0
       (/
        (fma
         (* t_2 (* k (tan k)))
         (* t_m k)
         (* (* t_m (* t_m (* t_m 2.0))) (* t_2 (tan k))))
        l))
      (/
       2.0
       (/
        (*
         (*
          t_m
          (* (* (* t_m (tan k)) (fma k (/ k (* t_m t_m)) 2.0)) (/ 1.0 l)))
         (* 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 t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 2.3e+44) {
		tmp = 2.0 / (fma((t_2 * (k * tan(k))), (t_m * k), ((t_m * (t_m * (t_m * 2.0))) * (t_2 * tan(k)))) / l);
	} else {
		tmp = 2.0 / (((t_m * (((t_m * tan(k)) * fma(k, (k / (t_m * t_m)), 2.0)) * (1.0 / l))) * (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)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 2.3e+44)
		tmp = Float64(2.0 / Float64(fma(Float64(t_2 * Float64(k * tan(k))), Float64(t_m * k), Float64(Float64(t_m * Float64(t_m * Float64(t_m * 2.0))) * Float64(t_2 * tan(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m * tan(k)) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(1.0 / l))) * 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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.3e+44], N[(2.0 / N[(N[(N[(t$95$2 * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision] + N[(N[(t$95$m * N[(t$95$m * N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.30000000000000004e44

   1. Initial program 51.7%

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied egg-rr 85.4%

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

   11. Applied egg-rr 87.7%

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

   if 2.30000000000000004e44 < t

   1. Initial program 61.5%

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

   4. Applied egg-rr 69.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied egg-rr 88.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right)} \cdot \left(t \cdot \sin k\right)}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{\sin k}{\ell} \cdot \left(k \cdot \tan k\right), t \cdot k, \left(t \cdot \left(t \cdot \left(t \cdot 2\right)\right)\right) \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t \cdot \sin k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_m \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{2 \cdot \ell}{t\_m \cdot \mathsf{fma}\left(k, t\_2 \cdot \left(k \cdot \tan k\right), \left(t\_m \cdot 2\right) \cdot \left(t\_2 \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \left(\left(t\_3 \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t\_m \cdot \sin k\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)) (t_3 (* t_m (tan k))))
   (*
    t_s
    (if (<= t_m 1.02e+49)
      (/
       (* 2.0 l)
       (* t_m (fma k (* t_2 (* k (tan k))) (* (* t_m 2.0) (* t_2 t_3)))))
      (/
       2.0
       (/
        (*
         (* t_m (* (* t_3 (fma k (/ k (* t_m t_m)) 2.0)) (/ 1.0 l)))
         (* 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 t_2 = sin(k) / l;
	double t_3 = t_m * tan(k);
	double tmp;
	if (t_m <= 1.02e+49) {
		tmp = (2.0 * l) / (t_m * fma(k, (t_2 * (k * tan(k))), ((t_m * 2.0) * (t_2 * t_3))));
	} else {
		tmp = 2.0 / (((t_m * ((t_3 * fma(k, (k / (t_m * t_m)), 2.0)) * (1.0 / l))) * (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)
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(t_m * tan(k))
	tmp = 0.0
	if (t_m <= 1.02e+49)
		tmp = Float64(Float64(2.0 * l) / Float64(t_m * fma(k, Float64(t_2 * Float64(k * tan(k))), Float64(Float64(t_m * 2.0) * Float64(t_2 * t_3)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_3 * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(1.0 / l))) * 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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.02e+49], N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[(k * N[(t$95$2 * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(t$95$3 * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.02e49

   1. Initial program 51.9%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied egg-rr 85.6%

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

   11. Applied egg-rr 88.0%

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

   if 1.02e49 < t

   1. Initial program 61.1%

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

   4. Applied egg-rr 69.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied egg-rr 87.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right)} \cdot \left(t \cdot \sin k\right)}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\frac{2 \cdot \ell}{t \cdot \mathsf{fma}\left(k, \frac{\sin k}{\ell} \cdot \left(k \cdot \tan k\right), \left(t \cdot 2\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t \cdot \tan k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t \cdot \sin k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_m \cdot \tan k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\ell \cdot \frac{2}{t\_m \cdot \mathsf{fma}\left(k, t\_2 \cdot \left(k \cdot \tan k\right), \left(t\_m \cdot 2\right) \cdot \left(t\_2 \cdot t\_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \left(\left(t\_3 \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t\_m \cdot \sin k\right)}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)) (t_3 (* t_m (tan k))))
   (*
    t_s
    (if (<= t_m 1.02e+49)
      (*
       l
       (/
        2.0
        (* t_m (fma k (* t_2 (* k (tan k))) (* (* t_m 2.0) (* t_2 t_3))))))
      (/
       2.0
       (/
        (*
         (* t_m (* (* t_3 (fma k (/ k (* t_m t_m)) 2.0)) (/ 1.0 l)))
         (* 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 t_2 = sin(k) / l;
	double t_3 = t_m * tan(k);
	double tmp;
	if (t_m <= 1.02e+49) {
		tmp = l * (2.0 / (t_m * fma(k, (t_2 * (k * tan(k))), ((t_m * 2.0) * (t_2 * t_3)))));
	} else {
		tmp = 2.0 / (((t_m * ((t_3 * fma(k, (k / (t_m * t_m)), 2.0)) * (1.0 / l))) * (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)
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(t_m * tan(k))
	tmp = 0.0
	if (t_m <= 1.02e+49)
		tmp = Float64(l * Float64(2.0 / Float64(t_m * fma(k, Float64(t_2 * Float64(k * tan(k))), Float64(Float64(t_m * 2.0) * Float64(t_2 * t_3))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_3 * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(1.0 / l))) * 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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.02e+49], N[(l * N[(2.0 / N[(t$95$m * N[(k * N[(t$95$2 * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(t$95$3 * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.02e49

   1. Initial program 51.9%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied egg-rr 85.6%

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

   11. Applied egg-rr 88.0%

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

   if 1.02e49 < t

   1. Initial program 61.1%

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

   4. Applied egg-rr 69.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied egg-rr 87.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right)} \cdot \left(t \cdot \sin k\right)}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{+49}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \mathsf{fma}\left(k, \frac{\sin k}{\ell} \cdot \left(k \cdot \tan k\right), \left(t \cdot 2\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t \cdot \tan k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t \cdot \sin k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \frac{2}{\mathsf{fma}\left(k, t\_2 \cdot \left(k \cdot \tan k\right), \left(t\_m \cdot 2\right) \cdot \left(t\_2 \cdot \left(t\_m \cdot \tan k\right)\right)\right) \cdot \frac{t\_m}{\ell}} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)))
   (*
    t_s
    (/
     2.0
     (*
      (fma k (* t_2 (* k (tan k))) (* (* t_m 2.0) (* t_2 (* t_m (tan k)))))
      (/ t_m 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 t_2 = sin(k) / l;
	return t_s * (2.0 / (fma(k, (t_2 * (k * tan(k))), ((t_m * 2.0) * (t_2 * (t_m * tan(k))))) * (t_m / l)));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	return Float64(t_s * Float64(2.0 / Float64(fma(k, Float64(t_2 * Float64(k * tan(k))), Float64(Float64(t_m * 2.0) * Float64(t_2 * Float64(t_m * tan(k))))) * Float64(t_m / l))))
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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * N[(2.0 / N[(N[(k * N[(t$95$2 * N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(t$95$2 * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.8%

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

4. Applied egg-rr 56.9%

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

5. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

7. Simplified 75.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

9. Applied egg-rr 82.9%

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

11. Applied egg-rr 88.4%

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

12. Final simplification 88.4%

    \[\leadsto \frac{2}{\mathsf{fma}\left(k, \frac{\sin k}{\ell} \cdot \left(k \cdot \tan k\right), \left(t \cdot 2\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t \cdot \tan k\right)\right)\right) \cdot \frac{t}{\ell}} \]
13. Add Preprocessing

Alternative 5: 87.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), k, 2 \cdot \left(k \cdot \left(k \cdot \frac{t\_m \cdot t\_m}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \left(\left(\left(t\_m \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t\_m \cdot \sin k\right)}{\ell}}\\ \end{array} \end{array} \]

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 5.8e-93)
    (/
     2.0
     (/
      (*
       t_m
       (fma
        (* k (* (/ (sin k) l) (tan k)))
        k
        (* 2.0 (* k (* k (/ (* t_m t_m) l))))))
      l))
    (/
     2.0
     (/
      (*
       (* t_m (* (* (* t_m (tan k)) (fma k (/ k (* t_m t_m)) 2.0)) (/ 1.0 l)))
       (* 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 <= 5.8e-93) {
		tmp = 2.0 / ((t_m * fma((k * ((sin(k) / l) * tan(k))), k, (2.0 * (k * (k * ((t_m * t_m) / l)))))) / l);
	} else {
		tmp = 2.0 / (((t_m * (((t_m * tan(k)) * fma(k, (k / (t_m * t_m)), 2.0)) * (1.0 / l))) * (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 <= 5.8e-93)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(Float64(k * Float64(Float64(sin(k) / l) * tan(k))), k, Float64(2.0 * Float64(k * Float64(k * Float64(Float64(t_m * t_m) / l)))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m * tan(k)) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(1.0 / l))) * 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, 5.8e-93], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(2.0 * N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999997e-93

   1. Initial program 46.6%

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

   4. Applied egg-rr 44.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 76.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied egg-rr 83.5%

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

   10. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 81.7

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

   12. Simplified 81.7%

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

   if 5.7999999999999997e-93 < t

   1. Initial program 65.4%

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

   4. Applied egg-rr 76.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied egg-rr 87.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right)} \cdot \left(t \cdot \sin k\right)}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), k, 2 \cdot \left(k \cdot \left(k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\left(\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(t \cdot \sin k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \mathsf{fma}\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), k, 2 \cdot \left(k \cdot \left(k \cdot \frac{t\_m \cdot t\_m}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \sin k\right) \cdot \left(t\_m \cdot \frac{\left(t\_m \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 5.8e-93)
    (/
     2.0
     (/
      (*
       t_m
       (fma
        (* k (* (/ (sin k) l) (tan k)))
        k
        (* 2.0 (* k (* k (/ (* t_m t_m) l))))))
      l))
    (/
     2.0
     (/
      (*
       (* t_m (sin k))
       (* t_m (/ (* (* t_m (tan k)) (fma k (/ k (* t_m t_m)) 2.0)) l)))
      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 <= 5.8e-93) {
		tmp = 2.0 / ((t_m * fma((k * ((sin(k) / l) * tan(k))), k, (2.0 * (k * (k * ((t_m * t_m) / l)))))) / l);
	} else {
		tmp = 2.0 / (((t_m * sin(k)) * (t_m * (((t_m * tan(k)) * fma(k, (k / (t_m * t_m)), 2.0)) / l))) / 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 <= 5.8e-93)
		tmp = Float64(2.0 / Float64(Float64(t_m * fma(Float64(k * Float64(Float64(sin(k) / l) * tan(k))), k, Float64(2.0 * Float64(k * Float64(k * Float64(Float64(t_m * t_m) / l)))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(t_m * Float64(Float64(Float64(t_m * tan(k)) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) / l))) / 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, 5.8e-93], N[(2.0 / N[(N[(t$95$m * N[(N[(k * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(2.0 * N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999997e-93

   1. Initial program 46.6%

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

   4. Applied egg-rr 44.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 76.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied egg-rr 83.5%

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

   10. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 81.7

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

   12. Simplified 81.7%

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

   if 5.7999999999999997e-93 < t

   1. Initial program 65.4%

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

   4. Applied egg-rr 76.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 80.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.1

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

   8. Applied egg-rr 87.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \mathsf{fma}\left(k \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right), k, 2 \cdot \left(k \cdot \left(k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \left(t \cdot \frac{\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right) \cdot \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \sin k\right) \cdot \left(t\_m \cdot \frac{\left(t\_m \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 4.9e-52)
    (/
     2.0
     (* (* t_m (* (/ (sin k) l) (tan k))) (/ (fma 2.0 (* t_m t_m) (* k k)) l)))
    (/
     2.0
     (/
      (*
       (* t_m (sin k))
       (* t_m (/ (* (* t_m (tan k)) (fma k (/ k (* t_m t_m)) 2.0)) l)))
      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 <= 4.9e-52) {
		tmp = 2.0 / ((t_m * ((sin(k) / l) * tan(k))) * (fma(2.0, (t_m * t_m), (k * k)) / l));
	} else {
		tmp = 2.0 / (((t_m * sin(k)) * (t_m * (((t_m * tan(k)) * fma(k, (k / (t_m * t_m)), 2.0)) / l))) / 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 <= 4.9e-52)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(sin(k) / l) * tan(k))) * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(t_m * Float64(Float64(Float64(t_m * tan(k)) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) / l))) / 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, 4.9e-52], N[(2.0 / N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[(N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.90000000000000019e-52

   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

   4. Applied egg-rr 48.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

   9. Applied egg-rr 80.9%

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

   if 4.90000000000000019e-52 < t

   1. Initial program 64.7%

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

   4. Applied egg-rr 73.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 78.6%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 86.3

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

   8. Applied egg-rr 86.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right) \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \left(t \cdot \frac{\left(t \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right) \cdot \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\sin k \cdot \frac{\left(t\_m \cdot t\_m\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 4e-46)
    (/
     2.0
     (* (* t_m (* (/ (sin k) l) (tan k))) (/ (fma 2.0 (* t_m t_m) (* k k)) l)))
    (/
     2.0
     (/
      (*
       t_m
       (*
        (sin k)
        (/ (* (* t_m t_m) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))) l)))
      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 <= 4e-46) {
		tmp = 2.0 / ((t_m * ((sin(k) / l) * tan(k))) * (fma(2.0, (t_m * t_m), (k * k)) / l));
	} else {
		tmp = 2.0 / ((t_m * (sin(k) * (((t_m * t_m) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))) / l))) / 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 <= 4e-46)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(sin(k) / l) * tan(k))) * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(sin(k) * Float64(Float64(Float64(t_m * t_m) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))) / l))) / 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, 4e-46], N[(2.0 / N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.00000000000000009e-46

   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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

   9. Applied egg-rr 80.5%

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

   if 4.00000000000000009e-46 < t

   1. Initial program 65.5%

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

   4. Applied egg-rr 74.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

   6. Applied egg-rr 79.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right) \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\sin k \cdot \frac{\left(t \cdot t\right) \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right) \cdot \frac{\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \sin k\right) \cdot \frac{k \cdot \left(2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 2.9e+38)
    (/
     2.0
     (* (* t_m (* (/ (sin k) l) (tan k))) (/ (fma 2.0 (* t_m t_m) (* k k)) l)))
    (/ 2.0 (/ (* (* t_m (sin k)) (/ (* k (* 2.0 (* t_m t_m))) l)) 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 <= 2.9e+38) {
		tmp = 2.0 / ((t_m * ((sin(k) / l) * tan(k))) * (fma(2.0, (t_m * t_m), (k * k)) / l));
	} else {
		tmp = 2.0 / (((t_m * sin(k)) * ((k * (2.0 * (t_m * t_m))) / l)) / 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 <= 2.9e+38)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(sin(k) / l) * tan(k))) * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(Float64(k * Float64(2.0 * Float64(t_m * t_m))) / l)) / 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, 2.9e+38], N[(2.0 / N[(N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.90000000000000007e38

   1. Initial program 51.4%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

   9. Applied egg-rr 81.6%

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

   if 2.90000000000000007e38 < t

   1. Initial program 62.2%

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

   4. Applied egg-rr 67.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 75.1%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 75.1

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

   9. Simplified 75.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot \tan k\right)\right) \cdot \frac{\mathsf{fma}\left(2, t \cdot t, k \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \frac{k \cdot \left(2 \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\left(t\_m \cdot \sin k\right) \cdot \frac{k \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot \left(k \cdot \mathsf{fma}\left(t\_m \cdot t\_m, 0.6666666666666666, 1\right)\right)\right)}{\ell}}{\ell}} \end{array} \]

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
  (/
   2.0
   (/
    (*
     (* t_m (sin k))
     (/
      (*
       k
       (fma
        2.0
        (* t_m t_m)
        (* k (* k (fma (* t_m t_m) 0.6666666666666666 1.0)))))
      l))
    l))))

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 * (2.0 / (((t_m * sin(k)) * ((k * fma(2.0, (t_m * t_m), (k * (k * fma((t_m * t_m), 0.6666666666666666, 1.0))))) / l)) / l));
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(Float64(k * fma(2.0, Float64(t_m * t_m), Float64(k * Float64(k * fma(Float64(t_m * t_m), 0.6666666666666666, 1.0))))) / l)) / l)))
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[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.6666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

4. Applied egg-rr 56.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. distribute-lft-in N/A

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

   11. rgt-mult-inverse N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

       \[\leadsto \frac{2}{\frac{\frac{k \cdot \mathsf{fma}\left(2, t \cdot t, \left(\mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{2}{3}, 1\right) \cdot k\right) \cdot k\right)}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \]

   14. lower-*.f64 72.4

       \[\leadsto \frac{2}{\frac{\frac{k \cdot \mathsf{fma}\left(2, t \cdot t, \left(\mathsf{fma}\left(\color{blue}{t \cdot t}, 0.6666666666666666, 1\right) \cdot k\right) \cdot k\right)}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \]

9. Simplified 72.4%

   \[\leadsto \frac{2}{\frac{\frac{\color{blue}{k \cdot \mathsf{fma}\left(2, t \cdot t, \left(\mathsf{fma}\left(t \cdot t, 0.6666666666666666, 1\right) \cdot k\right) \cdot k\right)}}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \]

10. Final simplification 72.4%

    \[\leadsto \frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \frac{k \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot \left(k \cdot \mathsf{fma}\left(t \cdot t, 0.6666666666666666, 1\right)\right)\right)}{\ell}}{\ell}} \]
11. Add Preprocessing

Alternative 11: 70.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4400000000:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right)\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot \sin k\right) \cdot \frac{k \cdot \left(2 \cdot \left(t\_m \cdot t\_m\right)\right)}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 4400000000.0)
    (/
     2.0
     (/
      (*
       t_m
       (*
        (fma 2.0 (* t_m t_m) (* k k))
        (* k (* k (fma (/ (* k k) l) 0.16666666666666666 (/ 1.0 l))))))
      l))
    (/ 2.0 (/ (* (* t_m (sin k)) (/ (* k (* 2.0 (* t_m t_m))) l)) 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 <= 4400000000.0) {
		tmp = 2.0 / ((t_m * (fma(2.0, (t_m * t_m), (k * k)) * (k * (k * fma(((k * k) / l), 0.16666666666666666, (1.0 / l)))))) / l);
	} else {
		tmp = 2.0 / (((t_m * sin(k)) * ((k * (2.0 * (t_m * t_m))) / l)) / 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 <= 4400000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) * Float64(k * Float64(k * fma(Float64(Float64(k * k) / l), 0.16666666666666666, Float64(1.0 / l)))))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(Float64(k * Float64(2.0 * Float64(t_m * t_m))) / l)) / 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, 4400000000.0], N[(2.0 / N[(N[(t$95$m * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.4e9

   1. Initial program 50.6%

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

   4. Applied egg-rr 52.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.1%

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

   8. Taylor expanded in k around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6} \cdot {k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6}}{\ell} \cdot {k}^{2}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\frac{\color{blue}{\frac{1}{6} \cdot 1}}{\ell} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot \frac{1}{\ell}\right)} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{\ell}\right) \cdot {k}^{2} + \frac{1}{\ell}\right)\right)}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6} \cdot 1}{\ell}} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\frac{\color{blue}{\frac{1}{6}}}{\ell} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6} \cdot {k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{{k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{1}{6}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2}}{\ell}, \frac{1}{6}, \frac{1}{\ell}\right)}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \mathsf{fma}\left(\color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{6}, \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{6}, \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{6}, \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      18. lower-/.f64 70.3

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

   10. Simplified 70.3%

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

   if 4.4e9 < t

   1. Initial program 63.4%

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

   4. Applied egg-rr 71.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 73.3

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

   9. Simplified 73.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4400000000:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\mathsf{fma}\left(2, t \cdot t, k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right)\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \frac{k \cdot \left(2 \cdot \left(t \cdot t\right)\right)}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right)\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

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 2.1e+41)
    (/
     2.0
     (/
      (*
       t_m
       (*
        (fma 2.0 (* t_m t_m) (* k k))
        (* k (* k (fma (/ (* k k) l) 0.16666666666666666 (/ 1.0 l))))))
      l))
    (* l (/ l (* (* t_m k) (* k (* t_m 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 (t_m <= 2.1e+41) {
		tmp = 2.0 / ((t_m * (fma(2.0, (t_m * t_m), (k * k)) * (k * (k * fma(((k * k) / l), 0.16666666666666666, (1.0 / l)))))) / l);
	} else {
		tmp = l * (l / ((t_m * k) * (k * (t_m * 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 (t_m <= 2.1e+41)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) * Float64(k * Float64(k * fma(Float64(Float64(k * k) / l), 0.16666666666666666, Float64(1.0 / l)))))) / l));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(k * Float64(t_m * 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[t$95$m, 2.1e+41], N[(2.0 / N[(N[(t$95$m * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e41

   1. Initial program 51.4%

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 77.9%

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

   8. Taylor expanded in k around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\color{blue}{\left(k \cdot \left(k \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2}}{\ell} + \frac{1}{\ell}\right)\right)\right)} \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6} \cdot {k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6}}{\ell} \cdot {k}^{2}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\frac{\color{blue}{\frac{1}{6} \cdot 1}}{\ell} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot \frac{1}{\ell}\right)} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \color{blue}{\left(k \cdot \left(\left(\frac{1}{6} \cdot \frac{1}{\ell}\right) \cdot {k}^{2} + \frac{1}{\ell}\right)\right)}\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6} \cdot 1}{\ell}} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\frac{\color{blue}{\frac{1}{6}}}{\ell} \cdot {k}^{2} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{\frac{1}{6} \cdot {k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{1}{6} \cdot \frac{{k}^{2}}{\ell}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \left(\color{blue}{\frac{{k}^{2}}{\ell} \cdot \frac{1}{6}} + \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \color{blue}{\mathsf{fma}\left(\frac{{k}^{2}}{\ell}, \frac{1}{6}, \frac{1}{\ell}\right)}\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \mathsf{fma}\left(\color{blue}{\frac{{k}^{2}}{\ell}}, \frac{1}{6}, \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{6}, \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(\left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{\color{blue}{k \cdot k}}{\ell}, \frac{1}{6}, \frac{1}{\ell}\right)\right)\right) \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}{\ell}} \]

      18. lower-/.f64 69.9

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

   10. Simplified 69.9%

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

   if 2.1e41 < t

   1. Initial program 62.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 53.1

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

   5. Simplified 53.1%

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

   6. Taylor expanded in t around 0

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 60.4

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

   8. Simplified 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 66.4

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

   10. Applied egg-rr 66.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 73.3

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

   12. Applied egg-rr 73.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\mathsf{fma}\left(2, t \cdot t, k \cdot k\right) \cdot \left(k \cdot \left(k \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell}, 0.16666666666666666, \frac{1}{\ell}\right)\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-97}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t\_m \cdot k}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(\mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right)}{\ell}}\\ \end{array} \end{array} \]

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 4.6e-97)
    (* l (/ (/ (/ l k) (* t_m k)) (* t_m t_m)))
    (/ 2.0 (/ (* t_m (* (fma 2.0 (* t_m t_m) (* k k)) (/ (* k k) l))) 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 (k <= 4.6e-97) {
		tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m));
	} else {
		tmp = 2.0 / ((t_m * (fma(2.0, (t_m * t_m), (k * k)) * ((k * k) / l))) / 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 (k <= 4.6e-97)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / Float64(t_m * k)) / Float64(t_m * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(fma(2.0, Float64(t_m * t_m), Float64(k * k)) * Float64(Float64(k * k) / l))) / 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[k, 4.6e-97], N[(l * N[(N[(N[(l / k), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.59999999999999988e-97

   1. Initial program 58.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.0

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

   5. Simplified 52.0%

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

   6. Taylor expanded in t around 0

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.1

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

   8. Simplified 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 59.6

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

   10. Applied egg-rr 59.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 68.2

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

   12. Applied egg-rr 68.2%

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

   if 4.59999999999999988e-97 < k

   1. Initial program 45.2%

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

   4. Applied egg-rr 49.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

   7. Simplified 81.0%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 64.5

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

   10. Simplified 64.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-97}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\mathsf{fma}\left(2, t \cdot t, k \cdot k\right) \cdot \frac{k \cdot k}{\ell}\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t\_m \cdot k}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t\_m, \left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333, t\_m\right)}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

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 5e-20)
    (* l (/ (/ (/ l k) (* t_m k)) (* t_m t_m)))
    (/
     2.0
     (*
      (* k k)
      (*
       (* k k)
       (/ (fma t_m (* (* t_m t_m) 0.3333333333333333) t_m) (* l 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 (k <= 5e-20) {
		tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m));
	} else {
		tmp = 2.0 / ((k * k) * ((k * k) * (fma(t_m, ((t_m * t_m) * 0.3333333333333333), t_m) / (l * 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 (k <= 5e-20)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / Float64(t_m * k)) / Float64(t_m * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(k * k) * Float64(fma(t_m, Float64(Float64(t_m * t_m) * 0.3333333333333333), t_m) / Float64(l * 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[k, 5e-20], N[(l * N[(N[(N[(l / k), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + t$95$m), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999999e-20

   1. Initial program 58.5%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 54.4

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

   5. Simplified 54.4%

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

   6. Taylor expanded in t around 0

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

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

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 59.0

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 61.9

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 61.9%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

       8. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{k \cdot t}}{t \cdot t}} \cdot \ell \]

       9. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{k \cdot t}}{t \cdot t}} \cdot \ell \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k}}{k \cdot t}}}{t \cdot t} \cdot \ell \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k}}}{k \cdot t}}{t \cdot t} \cdot \ell \]

       12. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{\ell}{k}}{\color{blue}{t \cdot k}}}{t \cdot t} \cdot \ell \]

       13. lower-*.f64 69.7

           \[\leadsto \frac{\frac{\frac{\ell}{k}}{\color{blue}{t \cdot k}}}{t \cdot t} \cdot \ell \]

   12. Applied egg-rr 69.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t \cdot t}} \cdot \ell \]

   if 4.9999999999999999e-20 < k

   1. Initial program 42.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot 2} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{t}^{3} \cdot 2}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{{t}^{3} \cdot \frac{2}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({t}^{3}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot \left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{{t}^{2}}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot {t}^{2}}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \color{blue}{\frac{2}{{\ell}^{2}}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \frac{\color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot {k}^{2}}}{{\ell}^{2}}\right)} \]

      17. associate-/l* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}\right)} \]

   5. Simplified 48.5%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \frac{2}{\ell \cdot \ell} + \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3}\right) + t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{2}{\ell \cdot \ell} + \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3}\right) + t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\color{blue}{\ell \cdot \ell}} + \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3}\right) + t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\frac{2}{\ell \cdot \ell}} + \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3}\right) + t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{3}\right) + t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \frac{1}{3}\right)} + t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \color{blue}{\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right)} \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right) \cdot \frac{\color{blue}{k \cdot k}}{\ell \cdot \ell}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right) \cdot \frac{k \cdot k}{\color{blue}{\ell \cdot \ell}}\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right) \cdot \color{blue}{\frac{k \cdot k}{\ell \cdot \ell}}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell} + \color{blue}{\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left(\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell} + \left(t \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{\ell \cdot \ell}\right)}} \]

   7. Applied egg-rr 33.1%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right), \ell, \frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{2 \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell}\right)}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \ell}}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot \left(t + \frac{1}{3} \cdot {t}^{3}\right)}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \frac{t + \frac{1}{3} \cdot {t}^{3}}{{\ell}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left({k}^{2} \cdot \frac{t + \frac{1}{3} \cdot {t}^{3}}{{\ell}^{2}}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t + \frac{1}{3} \cdot {t}^{3}}{{\ell}^{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \frac{t + \frac{1}{3} \cdot {t}^{3}}{{\ell}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\frac{t + \frac{1}{3} \cdot {t}^{3}}{{\ell}^{2}}}\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{\frac{1}{3} \cdot {t}^{3} + t}}{{\ell}^{2}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{{t}^{3} \cdot \frac{1}{3}} + t}{{\ell}^{2}}\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{1}{3} + t}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot \frac{1}{3} + t}{{\ell}^{2}}\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \left({t}^{2} \cdot \frac{1}{3}\right)} + t}{{\ell}^{2}}\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(t, {t}^{2} \cdot \frac{1}{3}, t\right)}}{{\ell}^{2}}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \color{blue}{{t}^{2} \cdot \frac{1}{3}}, t\right)}{{\ell}^{2}}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{3}, t\right)}{{\ell}^{2}}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{3}, t\right)}{{\ell}^{2}}\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot \frac{1}{3}, t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

      16. lower-*.f64 53.9

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

   10. Simplified 53.9%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right)}{\ell \cdot \ell}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot \frac{\mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right)}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

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 4.6e-69)
    (/ 2.0 (* (* k k) (/ (* t_m (* k k)) (* l l))))
    (if (<= t_m 5.1e+72)
      (* (/ l k) (/ l (* k (* t_m (* t_m t_m)))))
      (* l (/ l (* (* t_m k) (* k (* t_m 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 (t_m <= 4.6e-69) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else if (t_m <= 5.1e+72) {
		tmp = (l / k) * (l / (k * (t_m * (t_m * t_m))));
	} else {
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))));
	}
	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 (t_m <= 4.6d-69) then
        tmp = 2.0d0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
    else if (t_m <= 5.1d+72) then
        tmp = (l / k) * (l / (k * (t_m * (t_m * t_m))))
    else
        tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))))
    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 (t_m <= 4.6e-69) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else if (t_m <= 5.1e+72) {
		tmp = (l / k) * (l / (k * (t_m * (t_m * t_m))));
	} else {
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))));
	}
	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 t_m <= 4.6e-69:
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
	elif t_m <= 5.1e+72:
		tmp = (l / k) * (l / (k * (t_m * (t_m * t_m))))
	else:
		tmp = l * (l / ((t_m * k) * (k * (t_m * 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 (t_m <= 4.6e-69)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * l))));
	elseif (t_m <= 5.1e+72)
		tmp = Float64(Float64(l / k) * Float64(l / Float64(k * Float64(t_m * Float64(t_m * t_m)))));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(k * Float64(t_m * t_m)))));
	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 (t_m <= 4.6e-69)
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	elseif (t_m <= 5.1e+72)
		tmp = (l / k) * (l / (k * (t_m * (t_m * t_m))));
	else
		tmp = l * (l / ((t_m * k) * (k * (t_m * t_m))));
	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[t$95$m, 4.6e-69], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.1e+72], N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.6000000000000001e-69

   1. Initial program 47.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot 2} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{t}^{3} \cdot 2}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{{t}^{3} \cdot \frac{2}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({t}^{3}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot \left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{{t}^{2}}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot {t}^{2}}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \color{blue}{\frac{2}{{\ell}^{2}}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \frac{\color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot {k}^{2}}}{{\ell}^{2}}\right)} \]

      17. associate-/l* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}\right)} \]

   5. Simplified 51.4%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      6. lower-*.f64 55.0

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 55.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}} \]

   if 4.6000000000000001e-69 < t < 5.09999999999999977e72

   1. Initial program 71.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot {k}^{2}\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)} \cdot {k}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 52.6

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 52.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.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. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 54.7

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k}} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \frac{\ell}{k} \]

      8. lower-/.f64 66.0

         \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \color{blue}{\frac{\ell}{k}} \]

   10. Applied egg-rr 66.0%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{k}} \]

   if 5.09999999999999977e72 < t

   1. Initial program 59.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot {k}^{2}\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)} \cdot {k}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 55.3

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 55.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.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. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 61.7

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 66.7%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. *-commutative N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

       4. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right) \cdot k} \cdot \ell \]

       6. associate-*r* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \ell \]

       7. associate-*l* N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

       9. *-commutative N/A

          \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \ell \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \ell \]

       11. lower-*.f64 74.6

           \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

   12. Applied egg-rr 74.6%

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t\_m \cdot k}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

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 9.5e+17)
    (* l (/ (/ (/ l k) (* t_m k)) (* t_m t_m)))
    (/ 2.0 (* (* k k) (/ (* t_m (* k k)) (* l 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 (k <= 9.5e+17) {
		tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m));
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	}
	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 <= 9.5d+17) then
        tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m))
    else
        tmp = 2.0d0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
    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 <= 9.5e+17) {
		tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m));
	} else {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	}
	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 <= 9.5e+17:
		tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m))
	else:
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * 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 (k <= 9.5e+17)
		tmp = Float64(l * Float64(Float64(Float64(l / k) / Float64(t_m * k)) / Float64(t_m * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * l))));
	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 <= 9.5e+17)
		tmp = l * (((l / k) / (t_m * k)) / (t_m * t_m));
	else
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	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, 9.5e+17], N[(l * N[(N[(N[(l / k), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.5e17

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot {k}^{2}\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)} \cdot {k}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

      16. unpow2 N/A

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

      17. lower-*.f64 55.8

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Simplified 55.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   7. Step-by-step derivation

      1. lower-/.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. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      12. lower-*.f64 60.2

          \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   8. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

      8. lower-/.f64 63.5

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

   10. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

       4. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{\color{blue}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k}}{k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}} \cdot \ell \]

       7. associate-*r* N/A

          \[\leadsto \frac{\frac{\ell}{k}}{\color{blue}{\left(k \cdot t\right) \cdot \left(t \cdot t\right)}} \cdot \ell \]

       8. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{k \cdot t}}{t \cdot t}} \cdot \ell \]

       9. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{k \cdot t}}{t \cdot t}} \cdot \ell \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k}}{k \cdot t}}}{t \cdot t} \cdot \ell \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k}}}{k \cdot t}}{t \cdot t} \cdot \ell \]

       12. *-commutative N/A

           \[\leadsto \frac{\frac{\frac{\ell}{k}}{\color{blue}{t \cdot k}}}{t \cdot t} \cdot \ell \]

       13. lower-*.f64 71.0

           \[\leadsto \frac{\frac{\frac{\ell}{k}}{\color{blue}{t \cdot k}}}{t \cdot t} \cdot \ell \]

   12. Applied egg-rr 71.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t \cdot t}} \cdot \ell \]

   if 9.5e17 < k

   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. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot 2} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{\frac{{t}^{3} \cdot 2}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      6. associate-/l* N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\color{blue}{{t}^{3} \cdot \frac{2}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left({t}^{3}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}} \]

      8. cube-mult N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot \left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{{t}^{2}}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\color{blue}{t \cdot {t}^{2}}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \color{blue}{\left(t \cdot t\right)}, \frac{2}{{\ell}^{2}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \color{blue}{\frac{2}{{\ell}^{2}}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\color{blue}{\ell \cdot \ell}}, \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \frac{\color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot {k}^{2}}}{{\ell}^{2}}\right)} \]

      17. associate-/l* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}\right)} \]

   5. Simplified 43.7%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(t \cdot \left(t \cdot t\right), \frac{2}{\ell \cdot \ell}, \mathsf{fma}\left(t, \left(t \cdot t\right) \cdot 0.3333333333333333, t\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{{\ell}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      6. lower-*.f64 49.8

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \frac{\left(k \cdot k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   8. Simplified 49.8%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{k}}{t \cdot k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\right) \end{array} \]

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 k) (* k (* t_m 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) {
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
}

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 * k) * (k * (t_m * t_m)))))
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 * k) * (k * (t_m * t_m)))));
}

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 * k) * (k * (t_m * t_m)))))

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(Float64(t_m * k) * Float64(k * Float64(t_m * t_m))))))
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 * k) * (k * (t_m * t_m)))));
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[(N[(t$95$m * k), $MachinePrecision] * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot {k}^{2}\right)}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)} \cdot {k}^{2}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   8. cube-mult N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

   16. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   17. lower-*.f64 51.0

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 51.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.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. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 54.3

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 57.7

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 57.7%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \cdot \ell \]

    3. *-commutative N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(t \cdot \left(t \cdot t\right)\right) \cdot k\right)}} \cdot \ell \]

    4. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right) \cdot k}} \cdot \ell \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right) \cdot k} \cdot \ell \]

    6. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(t \cdot t\right)\right)} \cdot k} \cdot \ell \]

    7. associate-*l* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

    9. *-commutative N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \ell \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right)} \cdot \left(\left(t \cdot t\right) \cdot k\right)} \cdot \ell \]

    11. lower-*.f64 61.6

        \[\leadsto \frac{\ell}{\left(t \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

12. Applied egg-rr 61.6%

    \[\leadsto \frac{\ell}{\color{blue}{\left(t \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

13. Final simplification 61.6%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
14. Add Preprocessing

Alternative 18: 63.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

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 (* k (* t_m (* k (* t_m 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) {
	return t_s * (l * (l / (k * (t_m * (k * (t_m * t_m))))));
}

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 / (k * (t_m * (k * (t_m * t_m))))))
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 / (k * (t_m * (k * (t_m * t_m))))));
}

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 / (k * (t_m * (k * (t_m * t_m))))))

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(k * Float64(t_m * Float64(k * Float64(t_m * t_m)))))))
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 / (k * (t_m * (k * (t_m * t_m))))));
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[(k * N[(t$95$m * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot {k}^{2}\right)}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)} \cdot {k}^{2}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   8. cube-mult N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

   16. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   17. lower-*.f64 51.0

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 51.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.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. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 54.3

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 57.7

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 57.7%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(k \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

    3. associate-*r* N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right)}} \cdot \ell \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right)}} \cdot \ell \]

    5. lower-*.f64 61.6

       \[\leadsto \frac{\ell}{k \cdot \left(\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot t\right)} \cdot \ell \]

12. Applied egg-rr 61.6%

    \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(k \cdot \left(t \cdot t\right)\right) \cdot t\right)}} \cdot \ell \]

13. Final simplification 61.6%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(k \cdot \left(t \cdot t\right)\right)\right)} \]
14. Add Preprocessing

Alternative 19: 59.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\right) \end{array} \]

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 (* k (* k (* t_m (* t_m 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) {
	return t_s * (l * (l / (k * (k * (t_m * (t_m * t_m))))));
}

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 / (k * (k * (t_m * (t_m * t_m))))))
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 / (k * (k * (t_m * (t_m * t_m))))));
}

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 / (k * (k * (t_m * (t_m * t_m))))))

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(k * Float64(k * Float64(t_m * Float64(t_m * t_m)))))))
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 / (k * (k * (t_m * (t_m * t_m))))));
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[(k * N[(k * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{2 \cdot \frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{{\ell}^{2}}} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot {k}^{2}\right)}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot {k}^{2}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{{\ell}^{2}} \cdot 2\right)} \cdot {k}^{2}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{{\ell}^{2}}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   8. cube-mult N/A

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{{t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {t}^{2}}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \left(2 \cdot {k}^{2}\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]

   16. unpow2 N/A

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   17. lower-*.f64 51.0

       \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

5. Simplified 51.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(t \cdot t\right)}{\ell \cdot \ell} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}} \]

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
7. Step-by-step derivation

   1. lower-/.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. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \]

   8. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{{t}^{2}}\right)\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot {t}^{2}\right)}\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   12. lower-*.f64 54.3

       \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

8. Simplified 54.3%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \left(t \cdot t\right)\right)}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   5. associate-/l* N/A

      \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

   8. lower-/.f64 57.7

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}} \cdot \ell \]

10. Applied egg-rr 57.7%

    \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \cdot \ell} \]

11. Final simplification 57.7%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))