Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 93.4%

Time: 17.1s

Alternatives: 17

Speedup: 12.5×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-86}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{t\_m} \cdot \frac{k}{\ell}, 2 \cdot \frac{t\_m}{\ell}\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-86)
    (* l (/ (* 2.0 (* l (cos k))) (* (pow (sin k) 2.0) (* k (* t_m k)))))
    (/
     2.0
     (*
      (fma k (* (/ 1.0 t_m) (/ k l)) (* 2.0 (/ t_m l)))
      (* (tan k) (* t_m (/ (* t_m (sin k)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-86) {
		tmp = l * ((2.0 * (l * cos(k))) / (pow(sin(k), 2.0) * (k * (t_m * k))));
	} else {
		tmp = 2.0 / (fma(k, ((1.0 / t_m) * (k / l)), (2.0 * (t_m / l))) * (tan(k) * (t_m * ((t_m * sin(k)) / l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-86)
		tmp = Float64(l * Float64(Float64(2.0 * Float64(l * cos(k))) / Float64((sin(k) ^ 2.0) * Float64(k * Float64(t_m * k)))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(Float64(1.0 / t_m) * Float64(k / l)), Float64(2.0 * Float64(t_m / l))) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-86], N[(l * N[(N[(2.0 * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.1000000000000001e-86

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 35.2

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

   7. Applied rewrites 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 39.9%

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

   10. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-pow.f64 N/A

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

       12. lower-sin.f64 N/A

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

       13. unpow2 N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 80.7

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

   12. Applied rewrites 80.7%

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

   if 1.1000000000000001e-86 < t

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 65.7

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

   4. Applied rewrites 62.1%

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

   6. Applied rewrites 87.8%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 90.3

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

   9. Applied rewrites 90.3%

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

   11. Applied rewrites 93.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-86}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \left(\ell \cdot \cos k\right)}{{\sin k}^{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{t} \cdot \frac{k}{\ell}, 2 \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{t\_m} \cdot \frac{k}{\ell}, 2 \cdot \frac{t\_m}{\ell}\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.5e-96)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (/
     2.0
     (*
      (fma k (* (/ 1.0 t_m) (/ k l)) (* 2.0 (/ t_m l)))
      (* (tan k) (* t_m (/ (* t_m (sin k)) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.5e-96) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / (fma(k, ((1.0 / t_m) * (k / l)), (2.0 * (t_m / l))) * (tan(k) * (t_m * ((t_m * sin(k)) / l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.5e-96)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(Float64(1.0 / t_m) * Float64(k / l)), Float64(2.0 * Float64(t_m / l))) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.5e-96], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999993e-96

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if 9.4999999999999993e-96 < t

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 65.3

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

   4. Applied rewrites 61.8%

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 89.4

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

   9. Applied rewrites 89.4%

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

   11. Applied rewrites 92.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{1}{t} \cdot \frac{k}{\ell}, 2 \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t\_m}}{\ell}, 2 \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 9.5e-96)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (/
     2.0
     (*
      (* (tan k) (* t_m (/ (* t_m (sin k)) l)))
      (fma k (/ (/ k t_m) l) (* 2.0 (/ 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 tmp;
	if (t_m <= 9.5e-96) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * fma(k, ((k / t_m) / l), (2.0 * (t_m / 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 <= 9.5e-96)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))) * fma(k, Float64(Float64(k / t_m) / l), Float64(2.0 * Float64(t_m / 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, 9.5e-96], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / t$95$m), $MachinePrecision] / l), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999993e-96

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if 9.4999999999999993e-96 < t

   1. Initial program 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 65.3

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

   4. Applied rewrites 61.8%

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

   6. Applied rewrites 86.9%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 89.4

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

   9. Applied rewrites 89.4%

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

   11. Applied rewrites 92.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, \frac{\frac{k}{t}}{\ell}, 2 \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(t\_m \cdot \frac{t\_m \cdot t\_m}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot 2\right)}\\ \end{array} \end{array} \]

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 2.75e-77)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (if (<= t_m 2e+131)
      (/
       (* 2.0 l)
       (*
        (* t_m (/ (* t_m t_m) l))
        (* (sin k) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0)))))
      (/
       2.0
       (* (* (tan k) (* t_m (/ (* t_m (sin k)) l))) (* (/ t_m l) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-77) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else if (t_m <= 2e+131) {
		tmp = (2.0 * l) / ((t_m * ((t_m * t_m) / l)) * (sin(k) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * ((t_m / l) * 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-77)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	elseif (t_m <= 2e+131)
		tmp = Float64(Float64(2.0 * l) / Float64(Float64(t_m * Float64(Float64(t_m * t_m) / l)) * Float64(sin(k) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))) * Float64(Float64(t_m / l) * 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-77], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+131], N[(N[(2.0 * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.74999999999999999e-77

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if 2.74999999999999999e-77 < t < 1.9999999999999998e131

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 39.6

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

   4. Applied rewrites 39.6%

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

   6. Applied rewrites 73.4%

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

   if 1.9999999999999998e131 < t

   1. Initial program 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 61.1

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

   4. Applied rewrites 52.8%

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 95.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(t \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.75 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(t\_m \cdot \frac{t\_m \cdot t\_m}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot 2\right)}\\ \end{array} \end{array} \]

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 2.75e-77)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (if (<= t_m 2e+131)
      (*
       l
       (/
        2.0
        (*
         (* t_m (/ (* t_m t_m) l))
         (* (sin k) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))
      (/
       2.0
       (* (* (tan k) (* t_m (/ (* t_m (sin k)) l))) (* (/ t_m l) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.75e-77) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else if (t_m <= 2e+131) {
		tmp = l * (2.0 / ((t_m * ((t_m * t_m) / l)) * (sin(k) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0)))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * ((t_m / l) * 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.75e-77)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	elseif (t_m <= 2e+131)
		tmp = Float64(l * Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m * t_m) / l)) * Float64(sin(k) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))) * Float64(Float64(t_m / l) * 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.75e-77], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+131], N[(l * N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.74999999999999999e-77

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if 2.74999999999999999e-77 < t < 1.9999999999999998e131

   1. Initial program 73.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 39.6

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

   4. Applied rewrites 39.6%

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

   6. Applied rewrites 72.8%

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

   if 1.9999999999999998e131 < t

   1. Initial program 63.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 61.1

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

   4. Applied rewrites 52.8%

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 95.4%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(t \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 9.5e-88)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (/
     2.0
     (*
      (* t_m (/ (* t_m (sin k)) l))
      (* (/ t_m l) (* (tan k) (fma k (/ k (* t_m t_m)) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.5e-88) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / ((t_m * ((t_m * sin(k)) / l)) * ((t_m / l) * (tan(k) * fma(k, (k / (t_m * t_m)), 2.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.5e-88)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(t_m * sin(k)) / l)) * Float64(Float64(t_m / l) * Float64(tan(k) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.5e-88], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.5e-88

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if 9.5e-88 < t

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 65.7

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

   4. Applied rewrites 62.1%

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

   6. Applied rewrites 91.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot \ell}, 2 \cdot \frac{t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.5e-79)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (/
     2.0
     (*
      (* (tan k) (* t_m (/ (* t_m (sin k)) l)))
      (fma k (/ k (* t_m l)) (* 2.0 (/ 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 tmp;
	if (t_m <= 3.5e-79) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * fma(k, (k / (t_m * l)), (2.0 * (t_m / 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 <= 3.5e-79)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))) * fma(k, Float64(k / Float64(t_m * l)), Float64(2.0 * Float64(t_m / 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, 3.5e-79], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5000000000000003e-79

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if 3.5000000000000003e-79 < t

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 65.7

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

   4. Applied rewrites 62.1%

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

   6. Applied rewrites 87.8%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 90.3

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

   9. Applied rewrites 90.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot \ell}, 2 \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot 2\right)}\\ \end{array} \end{array} \]

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 7.5e+52)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (/ 2.0 (* (* (tan k) (* t_m (/ (* t_m (sin k)) l))) (* (/ t_m l) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e+52) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * ((t_m / l) * 2.0));
	}
	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 <= 7.5d+52) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))))
    else
        tmp = 2.0d0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * ((t_m / l) * 2.0d0))
    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 <= 7.5e+52) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (t_m * ((t_m * Math.sin(k)) / l))) * ((t_m / l) * 2.0));
	}
	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 <= 7.5e+52:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (k * (k * (t_m / (l * l))))))
	else:
		tmp = 2.0 / ((math.tan(k) * (t_m * ((t_m * math.sin(k)) / l))) * ((t_m / l) * 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e+52)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * sin(k)) / l))) * Float64(Float64(t_m / l) * 2.0)));
	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 <= 7.5e+52)
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	else
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * sin(k)) / l))) * ((t_m / l) * 2.0));
	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, 7.5e+52], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.49999999999999995e52

   1. Initial program 53.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.3

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

   7. Applied rewrites 70.3%

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

   if 7.49999999999999995e52 < t

   1. Initial program 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 65.0

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

   4. Applied rewrites 59.1%

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

   6. Applied rewrites 93.7%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 88.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.68 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t\_m}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot \ell}, 2 \cdot \frac{t\_m}{\ell}\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \left(k \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.68e-15)
    (/ 2.0 (* (sin k) (* (tan k) (* k (* k (/ t_m (* l l)))))))
    (/
     2.0
     (*
      (fma k (/ k (* t_m l)) (* 2.0 (/ t_m l)))
      (* (tan k) (* t_m (* 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 tmp;
	if (t_m <= 1.68e-15) {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * (k * (t_m / (l * l))))));
	} else {
		tmp = 2.0 / (fma(k, (k / (t_m * l)), (2.0 * (t_m / l))) * (tan(k) * (t_m * (k * (t_m / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.68e-15)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(k * Float64(t_m / Float64(l * l)))))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(t_m * l)), Float64(2.0 * Float64(t_m / l))) * Float64(tan(k) * Float64(t_m * Float64(k * Float64(t_m / l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.68e-15], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(k * N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(t$95$m * l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6800000000000001e-15

   1. Initial program 50.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 70.6

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

   7. Applied rewrites 70.6%

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

   if 1.6800000000000001e-15 < t

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 67.9

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

   4. Applied rewrites 63.5%

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

   6. Applied rewrites 91.0%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 93.9

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

   9. Applied rewrites 93.9%

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

   10. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 80.9

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

   12. Applied rewrites 80.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.68 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell \cdot \ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t \cdot \ell}, 2 \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-166}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \frac{t\_m \cdot k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.95e-166)
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k)))))
    (/
     2.0
     (*
      (* (/ t_m l) (fma k (/ k (* t_m t_m)) 2.0))
      (* (tan k) (* t_m (/ (* t_m k) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.95e-166) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else {
		tmp = 2.0 / (((t_m / l) * fma(k, (k / (t_m * t_m)), 2.0)) * (tan(k) * (t_m * ((t_m * k) / l))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.95e-166)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * fma(k, Float64(k / Float64(t_m * t_m)), 2.0)) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * k) / l)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.95e-166], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.95e-166

   1. Initial program 49.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 63.9%

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

   if 1.95e-166 < t

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 64.0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 76.3

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

   9. Applied rewrites 76.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-166}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{t \cdot k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot \ell}, 2 \cdot \frac{t\_m}{\ell}\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \left(k \cdot \frac{t\_m}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e-159)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (/
     2.0
     (*
      (fma k (/ k (* t_m l)) (* 2.0 (/ t_m l)))
      (* (tan k) (* t_m (* 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 tmp;
	if (k <= 9.5e-159) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = 2.0 / (fma(k, (k / (t_m * l)), (2.0 * (t_m / l))) * (tan(k) * (t_m * (k * (t_m / 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-159)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(t_m * l)), Float64(2.0 * Float64(t_m / l))) * Float64(tan(k) * Float64(t_m * Float64(k * Float64(t_m / 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, 9.5e-159], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(t$95$m * l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(k * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.4999999999999997e-159

   1. Initial program 56.1%

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

   3. Taylor expanded in k around 0

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 62.8%

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

   11. Applied rewrites 64.0%

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

   if 9.4999999999999997e-159 < k

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 58.4

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

   4. Applied rewrites 49.7%

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.5

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

   9. Applied rewrites 84.5%

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

   10. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 68.6

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

   12. Applied rewrites 68.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t \cdot \ell}, 2 \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \left(t \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right) \cdot \left(\left(t\_m \cdot k\right) \cdot \frac{k \cdot \left(t\_m \cdot t\_m\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-162)
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k)))))
    (if (<= t_m 1.8e+70)
      (/
       (* 2.0 l)
       (* (fma k (/ k (* t_m t_m)) 2.0) (* (* t_m k) (/ (* k (* t_m t_m)) l))))
      (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-162) {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	} else if (t_m <= 1.8e+70) {
		tmp = (2.0 * l) / (fma(k, (k / (t_m * t_m)), 2.0) * ((t_m * k) * ((k * (t_m * t_m)) / l)));
	} else {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-162)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	elseif (t_m <= 1.8e+70)
		tmp = Float64(Float64(2.0 * l) / Float64(fma(k, Float64(k / Float64(t_m * t_m)), 2.0) * Float64(Float64(t_m * k) * Float64(Float64(k * Float64(t_m * t_m)) / l))));
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	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, 4.5e-162], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.8e+70], N[(N[(2.0 * l), $MachinePrecision] / N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] * N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.50000000000000023e-162

   1. Initial program 49.9%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 63.9%

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

   if 4.50000000000000023e-162 < t < 1.8e70

   1. Initial program 67.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 58.9

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

   7. Applied rewrites 58.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 60.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 60.6

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

   11. Applied rewrites 65.3%

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

   if 1.8e70 < t

   1. Initial program 66.5%

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

   3. Taylor expanded in k around 0

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 72.7%

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

   9. Applied rewrites 77.0%

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

   11. Applied rewrites 79.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{2 \cdot \ell}{\mathsf{fma}\left(k, \frac{k}{t \cdot t}, 2\right) \cdot \left(\left(t \cdot k\right) \cdot \frac{k \cdot \left(t \cdot t\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t\_m \cdot \ell}, 2 \cdot \frac{t\_m}{\ell}\right) \cdot \frac{t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}{\ell}}\\ \end{array} \end{array} \]

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 9.5e-159)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (/
     2.0
     (*
      (fma k (/ k (* t_m l)) (* 2.0 (/ t_m l)))
      (/ (* t_m (* k (* t_m 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 (k <= 9.5e-159) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = 2.0 / (fma(k, (k / (t_m * l)), (2.0 * (t_m / l))) * ((t_m * (k * (t_m * 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 (k <= 9.5e-159)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64(k / Float64(t_m * l)), Float64(2.0 * Float64(t_m / l))) * Float64(Float64(t_m * Float64(k * Float64(t_m * 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[k, 9.5e-159], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(k / N[(t$95$m * l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.4999999999999997e-159

   1. Initial program 56.1%

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

   3. Taylor expanded in k around 0

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 60.4%

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

   9. Applied rewrites 62.8%

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

   11. Applied rewrites 64.0%

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

   if 9.4999999999999997e-159 < k

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. cube-mult N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 58.4

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

   4. Applied rewrites 49.7%

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*l/ N/A

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

      15. associate-*r/ N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.5

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

   9. Applied rewrites 84.5%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 67.6

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

   12. Applied rewrites 67.6%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{t \cdot \ell}, 2 \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-155)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-155) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-155) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-155) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5e-155:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5e-155)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5e-155)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5e-155], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999999e-155

   1. Initial program 55.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

   7. Applied rewrites 60.0%

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

   9. Applied rewrites 62.4%

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

   11. Applied rewrites 64.2%

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

   if 4.9999999999999999e-155 < k

   1. Initial program 56.3%

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

   3. Taylor expanded in k around 0

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

      1. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 65.5%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+27}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.2e+27)
    (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))
    (/ (* l l) (* t_m (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e+27) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.2d+27) then
        tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
    else
        tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e+27) {
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	} else {
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 8.2e+27:
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))))
	else:
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8.2e+27)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k)))));
	else
		tmp = Float64(Float64(l * l) / Float64(t_m * Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 8.2e+27)
		tmp = l * (l / (t_m * ((t_m * k) * (t_m * k))));
	else
		tmp = (l * l) / (t_m * (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 8.2e+27], N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.2000000000000005e27

   1. Initial program 57.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

      12. lower-*.f64 51.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.0%

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \color{blue}{\ell} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.0%

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot \ell \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

   if 8.2000000000000005e27 < k

   1. Initial program 52.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} \]

      9. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

      10. associate-*r/ N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}} \]

   4. Applied rewrites 44.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + \frac{k \cdot k}{t \cdot t}\right) \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. 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. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left({t}^{2} \cdot {k}^{2}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left({k}^{2} \cdot {t}^{2}\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left({k}^{2} \cdot {t}^{2}\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(t \cdot \left(t \cdot {k}^{2}\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(t \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

      16. lower-*.f64 56.3

          \[\leadsto \frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \]

   7. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+27}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t \cdot \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot \left(t\_m \cdot k\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* t_m (* (* t_m k) (* t_m k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(t_m * Float64(Float64(t_m * k) * Float64(t_m * k))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (t_m * ((t_m * k) * (t_m * k)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 51.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 59.3%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \color{blue}{\ell} \]
8. Step-by-step derivation

9. Applied rewrites 60.9%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot \ell \]
10. Step-by-step derivation

11. Applied rewrites 62.8%

    \[\leadsto \frac{\ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \cdot \ell \]

12. Final simplification 62.8%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)} \]
13. Add Preprocessing

Alternative 17: 63.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t\_m \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* k (* t_m (* t_m (* t_m k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(k * Float64(t_m * Float64(t_m * Float64(t_m * k)))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / (k * (t_m * (t_m * (t_m * k))))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(k * N[(t$95$m * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   6. cube-mult N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot {k}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot {k}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot {k}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot {k}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   12. lower-*.f64 51.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 59.3%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \color{blue}{\ell} \]
8. Step-by-step derivation

9. Applied rewrites 60.9%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot \ell \]

10. Final simplification 60.9%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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))))