Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 88.4%

Time: 17.5s

Alternatives: 16

Speedup: 12.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% 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: 88.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot \left(t\_m \cdot {\sin k}^{2}\right)\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 6.8e-81)
    (/ 2.0 (/ (/ (* k (* k (* t_m (pow (sin k) 2.0)))) (* l (cos k))) l))
    (/
     2.0
     (*
      (* (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l))) (tan k))
      (fma (/ k t_m) (/ k 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 <= 6.8e-81) {
		tmp = 2.0 / (((k * (k * (t_m * pow(sin(k), 2.0)))) / (l * cos(k))) / l);
	} else {
		tmp = 2.0 / (((t_m * (((t_m * sin(k)) / l) * (t_m / l))) * tan(k)) * fma((k / t_m), (k / 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 <= 6.8e-81)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * Float64(t_m * (sin(k) ^ 2.0)))) / Float64(l * cos(k))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l))) * tan(k)) * fma(Float64(k / t_m), Float64(k / 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, 6.8e-81], N[(2.0 / N[(N[(N[(k * N[(k * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.7999999999999997e-81

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 22.3%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if 6.7999999999999997e-81 < t

   1. Initial program 70.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. cube-mult N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 91.6

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

   6. Applied rewrites 91.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 91.6

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

   8. Applied rewrites 91.6%

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

Alternative 2: 86.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{+41}:\\ \;\;\;\;\ell \cdot \frac{\frac{2 \cdot \ell}{\sin k \cdot \left(t\_m \cdot \mathsf{fma}\left(2, t\_m \cdot t\_m, k \cdot k\right)\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m \cdot \sin k}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 6e+41)
    (*
     l
     (/
      (/ (* 2.0 l) (* (sin k) (* t_m (fma 2.0 (* t_m t_m) (* k k)))))
      (tan k)))
    (/
     2.0
     (*
      (* (* t_m (* (/ (* t_m (sin k)) l) (/ t_m l))) (tan k))
      (fma (/ k t_m) (/ k 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 <= 6e+41) {
		tmp = l * (((2.0 * l) / (sin(k) * (t_m * fma(2.0, (t_m * t_m), (k * k))))) / tan(k));
	} else {
		tmp = 2.0 / (((t_m * (((t_m * sin(k)) / l) * (t_m / l))) * tan(k)) * fma((k / t_m), (k / 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 <= 6e+41)
		tmp = Float64(l * Float64(Float64(Float64(2.0 * l) / Float64(sin(k) * Float64(t_m * fma(2.0, Float64(t_m * t_m), Float64(k * k))))) / tan(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m * sin(k)) / l) * Float64(t_m / l))) * tan(k)) * fma(Float64(k / t_m), Float64(k / 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, 6e+41], N[(l * N[(N[(N[(2.0 * l), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.9999999999999997e41

   1. Initial program 46.9%

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

      1. lift-/.f64 N/A

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

      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. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 59.5

          \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 59.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 37.7%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 80.4

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

   9. Applied rewrites 80.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

   11. Applied rewrites 82.6%

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

   if 5.9999999999999997e41 < t

   1. Initial program 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. cube-mult N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 90.4

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

   6. Applied rewrites 90.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 90.4

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

   8. Applied rewrites 90.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+41}:\\ \;\;\;\;\ell \cdot \frac{\frac{2 \cdot \ell}{\sin k \cdot \left(t \cdot \mathsf{fma}\left(2, t \cdot t, k \cdot k\right)\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))