Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 90.4%

Time: 17.7s

Alternatives: 21

Speedup: 12.5×

• 27.0% 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: 90.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(t\_m \cdot \mathsf{fma}\left(t\_2, 2 \cdot \left(t\_m \cdot t\_m\right), k \cdot \left(k \cdot t\_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t\_m \cdot \left(\left(t\_m \cdot t\_2\right) \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 1.95e+109)
      (/
       2.0
       (* (/ (tan k) l) (* t_m (fma t_2 (* 2.0 (* t_m t_m)) (* k (* k t_2))))))
      (/
       2.0
       (*
        (* (tan k) (* t_m (* (* t_m t_2) (/ t_m l))))
        (+ 1.0 (+ 1.0 (pow (/ 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 t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 1.95e+109) {
		tmp = 2.0 / ((tan(k) / l) * (t_m * fma(t_2, (2.0 * (t_m * t_m)), (k * (k * t_2)))));
	} else {
		tmp = 2.0 / ((tan(k) * (t_m * ((t_m * t_2) * (t_m / l)))) * (1.0 + (1.0 + pow((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)
	t_2 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 1.95e+109)
		tmp = Float64(2.0 / Float64(Float64(tan(k) / l) * Float64(t_m * fma(t_2, Float64(2.0 * Float64(t_m * t_m)), Float64(k * Float64(k * t_2))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_m * Float64(Float64(t_m * t_2) * Float64(t_m / l)))) * Float64(1.0 + Float64(1.0 + (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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e+109], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(t$95$2 * N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(k * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * t$95$2), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.95000000000000008e109

   1. Initial program 43.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Simplified 72.0%

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

      1. lift-sin.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 81.8

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

   7. Applied egg-rr 81.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

   9. Applied egg-rr 87.7%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. distribute-lft-in N/A

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

       7. lower-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 94.2

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

   11. Applied egg-rr 94.2%

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

   if 1.95000000000000008e109 < t

   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-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 79.0

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

   4. Applied egg-rr 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 95.4

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

   6. Applied egg-rr 95.4%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{+109}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \left(t \cdot \mathsf{fma}\left(\frac{\sin k}{\ell}, 2 \cdot \left(t \cdot t\right), k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(t \cdot \left(\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.1% accurate, 0.9× speedup

Math

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

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 (<=
       (/
        2.0
        (*
         (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
         (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))
       INFINITY)
    (* (/ l (* t_m k)) (/ l (* k (* t_m t_m))))
    (* (/ 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 ((2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))))) <= ((double) INFINITY)) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))))) <= Double.POSITIVE_INFINITY) {
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	} 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 (2.0 / ((1.0 + (1.0 + math.pow((k / t_m), 2.0))) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))))) <= math.inf:
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)))
	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 (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))) <= Inf)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(k * Float64(t_m * t_m))));
	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 ((2.0 / ((1.0 + (1.0 + ((k / t_m) ^ 2.0))) * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))))) <= Inf)
		tmp = (l / (t_m * k)) * (l / (k * (t_m * t_m)));
	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[N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(t$95$m * t$95$m), $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 (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 82.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   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 71.7

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

   5. Simplified 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 72.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.0

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

   7. Applied egg-rr 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 80.6

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

   9. Applied egg-rr 80.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 83.6

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

   11. Applied egg-rr 83.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

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

      1. 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 10.4

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

   5. Simplified 10.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 40.1

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

   7. Applied egg-rr 40.1%

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

4. Final simplification 69.1%

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

Reproduce

herbie shell --seed 2024218 
(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))))