Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.1% → 90.0%

Time: 18.7s

Alternatives: 18

Speedup: 12.5×

• 27.1% 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: 54.1% 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: 90.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_m \cdot \frac{\tan k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-142}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(k \cdot \left(k \cdot t\_2\right)\right)}\\ \mathbf{elif}\;t\_m \leq 10^{+102}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(t\_2 \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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_3 (* t_m (/ (tan k) l))))
   (*
    t_s
    (if (<= t_m 1e-142)
      (/ 2.0 (* t_3 (* k (* k t_2))))
      (if (<= t_m 1e+102)
        (/ 2.0 (* t_3 (* t_2 (fma k k (* 2.0 (* t_m t_m))))))
        (/
         2.0
         (*
          (* (tan k) (* (/ t_m l) (* t_m (/ (* t_m (sin k)) 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 t_3 = t_m * (tan(k) / l);
	double tmp;
	if (t_m <= 1e-142) {
		tmp = 2.0 / (t_3 * (k * (k * t_2)));
	} else if (t_m <= 1e+102) {
		tmp = 2.0 / (t_3 * (t_2 * fma(k, k, (2.0 * (t_m * t_m)))));
	} else {
		tmp = 2.0 / ((tan(k) * ((t_m / l) * (t_m * ((t_m * sin(k)) / 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)
	t_3 = Float64(t_m * Float64(tan(k) / l))
	tmp = 0.0
	if (t_m <= 1e-142)
		tmp = Float64(2.0 / Float64(t_3 * Float64(k * Float64(k * t_2))));
	elseif (t_m <= 1e+102)
		tmp = Float64(2.0 / Float64(t_3 * Float64(t_2 * fma(k, k, Float64(2.0 * Float64(t_m * t_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(t_m / l) * Float64(t_m * Float64(Float64(t_m * sin(k)) / 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]}, Block[{t$95$3 = N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-142], N[(2.0 / N[(t$95$3 * N[(k * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+102], N[(2.0 / N[(t$95$3 * N[(t$95$2 * N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / 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 3 regimes

2. if t < 1e-142

   1. Initial program 48.2%

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

   3. Taylor expanded in 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. Applied rewrites 71.2%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in k around inf

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-sin.f64 76.0

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

   12. Applied rewrites 76.0%

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

   if 1e-142 < t < 9.99999999999999977e101

   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. 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. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.5%

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

   if 9.99999999999999977e101 < t

   1. Initial program 60.0%

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

      1. lift-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. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.8

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

   6. Applied rewrites 92.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-142}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 10^{+102}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.9% accurate, 0.9× speedup

Math

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

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 (<=
       (*
        (+ 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 (* t_m (* t_m k))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (((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 / (t_m * (t_m * k)));
	} 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 (((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 / (t_m * (t_m * k)));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if ((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 / (t_m * (t_m * k)))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (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(t_m * Float64(t_m * k))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (((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 / (t_m * (t_m * k)));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[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], Infinity], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $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 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 80.8%

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

   3. Taylor expanded in k around 0

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

      1. 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 71.5

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

   5. Applied rewrites 71.5%

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

      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 73.0

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 80.6

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

   7. Applied rewrites 80.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 80.6

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

   9. Applied rewrites 80.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 87.7

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 87.7

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 87.7

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

   11. Applied rewrites 87.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

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

      1. 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 16.6

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

   5. Applied rewrites 16.6%

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

      1. 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. lower-*.f64 N/A

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

      11. lower-*.f64 42.8

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

   7. Applied rewrites 42.8%

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

4. Final simplification 72.6%

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

Alternative 3: 69.0% accurate, 0.9× speedup

Math

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

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

   1. Initial program 80.8%

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

   3. Taylor expanded in k around 0

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

      1. 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 71.5

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

   5. Applied rewrites 71.5%

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

      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 73.0

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 80.6

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

   7. Applied rewrites 80.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 84.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 84.4

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

   9. Applied rewrites 84.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in k around 0

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

      1. 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 16.6

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

   5. Applied rewrites 16.6%

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

      1. 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. lower-*.f64 N/A

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

      11. lower-*.f64 42.8

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

   7. Applied rewrites 42.8%

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

4. Final simplification 70.4%

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

Alternative 4: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t_3 := t\_m \cdot \frac{\tan k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-142}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(k \cdot \left(k \cdot t\_2\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(t\_2 \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

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_3 (* t_m (/ (tan k) l))))
   (*
    t_s
    (if (<= t_m 1e-142)
      (/ 2.0 (* t_3 (* k (* k t_2))))
      (if (<= t_m 1.05e+103)
        (/ 2.0 (* t_3 (* t_2 (fma k k (* 2.0 (* t_m t_m))))))
        (/
         2.0
         (*
          (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
          (* (tan k) (* t_m (* (/ t_m l) (/ (* t_m (sin k)) l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) / l;
	double t_3 = t_m * (tan(k) / l);
	double tmp;
	if (t_m <= 1e-142) {
		tmp = 2.0 / (t_3 * (k * (k * t_2)));
	} else if (t_m <= 1.05e+103) {
		tmp = 2.0 / (t_3 * (t_2 * fma(k, k, (2.0 * (t_m * t_m)))));
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * (t_m * ((t_m / l) * ((t_m * sin(k)) / l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) / l)
	t_3 = Float64(t_m * Float64(tan(k) / l))
	tmp = 0.0
	if (t_m <= 1e-142)
		tmp = Float64(2.0 / Float64(t_3 * Float64(k * Float64(k * t_2))));
	elseif (t_m <= 1.05e+103)
		tmp = Float64(2.0 / Float64(t_3 * Float64(t_2 * fma(k, k, Float64(2.0 * Float64(t_m * t_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * Float64(t_m * Float64(Float64(t_m / l) * 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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-142], N[(2.0 / N[(t$95$3 * N[(k * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+103], N[(2.0 / N[(t$95$3 * N[(t$95$2 * N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1e-142

   1. Initial program 48.2%

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

   3. Taylor expanded in 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. Applied rewrites 71.2%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in k around inf

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-sin.f64 76.0

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

   12. Applied rewrites 76.0%

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

   if 1e-142 < t < 1.0500000000000001e103

   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. 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. Applied rewrites 73.0%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 89.5%

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

   if 1.0500000000000001e103 < t

   1. Initial program 60.0%

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

      1. lift-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. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 90.9

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

   6. Applied rewrites 90.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-142}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \left(t\_m \cdot \sin k\right)}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.26e-142)
    (/ 2.0 (* (* t_m (/ (tan k) l)) (* k (* k (/ (sin k) 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 <= 1.26e-142) {
		tmp = 2.0 / ((t_m * (tan(k) / l)) * (k * (k * (sin(k) / 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 <= 1.26e-142)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(tan(k) / l)) * Float64(k * Float64(k * Float64(sin(k) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * 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, 1.26e-142], N[(2.0 / N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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 < 1.26000000000000007e-142

   1. Initial program 48.2%

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

   3. Taylor expanded in 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. Applied rewrites 71.2%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in k around inf

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-sin.f64 76.0

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

   12. Applied rewrites 76.0%

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

   if 1.26000000000000007e-142 < t

   1. Initial program 63.5%

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

   4. Applied rewrites 64.3%

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

   6. Applied rewrites 90.1%

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

Alternative 6: 79.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-104}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+152}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{t\_m \cdot \tan k}}{\sin k \cdot \mathsf{fma}\left(t\_m, t\_m \cdot 2, k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

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 (<= k 2.75e-104)
    (* (/ l (* t_m k)) (/ l (* t_m (* t_m k))))
    (if (<= k 9e+152)
      (*
       l
       (/
        (* l (/ 2.0 (* t_m (tan k))))
        (* (sin k) (fma t_m (* t_m 2.0) (* k k)))))
      (/ 2.0 (* (* t_m (/ (tan k) l)) (* k (* k (/ (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 (k <= 2.75e-104) {
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	} else if (k <= 9e+152) {
		tmp = l * ((l * (2.0 / (t_m * tan(k)))) / (sin(k) * fma(t_m, (t_m * 2.0), (k * k))));
	} else {
		tmp = 2.0 / ((t_m * (tan(k) / l)) * (k * (k * (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 (k <= 2.75e-104)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * Float64(t_m * k))));
	elseif (k <= 9e+152)
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / Float64(t_m * tan(k)))) / Float64(sin(k) * fma(t_m, Float64(t_m * 2.0), Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(tan(k) / l)) * Float64(k * Float64(k * Float64(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[k, 2.75e-104], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9e+152], N[(l * N[(N[(l * N[(2.0 / N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(t$95$m * N[(t$95$m * 2.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.7499999999999999e-104

   1. Initial program 56.6%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.4

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

   5. Applied rewrites 53.4%

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

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

      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 57.4

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 68.5

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

   9. Applied rewrites 68.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 77.0

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 77.0

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 77.0

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

   11. Applied rewrites 77.0%

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

   if 2.7499999999999999e-104 < k < 9.0000000000000002e152

   1. Initial program 48.4%

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

   3. Taylor expanded in 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. Applied rewrites 85.8%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 88.9

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

   7. Applied rewrites 88.9%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 96.4%

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

       1. lift-tan.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/r* N/A

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

       11. lift-*.f64 N/A

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

       12. lift-/.f64 N/A

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

       13. associate-*l/ N/A

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

   11. Applied rewrites 96.7%

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

   if 9.0000000000000002e152 < k

   1. Initial program 50.2%

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

   3. Taylor expanded in 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. Applied rewrites 65.6%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 65.6

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

   7. Applied rewrites 65.6%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 65.9%

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

   10. Taylor expanded in k around inf

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-sin.f64 81.4

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

   12. Applied rewrites 81.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-104}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+152}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{t \cdot \tan k}}{\sin k \cdot \mathsf{fma}\left(t, t \cdot 2, k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 1320000000:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(t\_m \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot t\_2\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (<= k 2e-140)
      (* (/ l (* t_m k)) (/ l (* t_m (* t_m k))))
      (if (<= k 1320000000.0)
        (/ 2.0 (* (* t_2 (fma k k (* 2.0 (* t_m t_m)))) (* t_m (/ k l))))
        (/ 2.0 (* (* t_m (/ (tan k) l)) (* k (* k t_2)))))))))

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 (k <= 2e-140) {
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	} else if (k <= 1320000000.0) {
		tmp = 2.0 / ((t_2 * fma(k, k, (2.0 * (t_m * t_m)))) * (t_m * (k / l)));
	} else {
		tmp = 2.0 / ((t_m * (tan(k) / l)) * (k * (k * t_2)));
	}
	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 (k <= 2e-140)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * Float64(t_m * k))));
	elseif (k <= 1320000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_2 * fma(k, k, Float64(2.0 * Float64(t_m * t_m)))) * Float64(t_m * Float64(k / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(tan(k) / l)) * Float64(k * Float64(k * t_2))));
	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[k, 2e-140], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1320000000.0], N[(2.0 / N[(N[(t$95$2 * N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2e-140

   1. Initial program 56.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

      1. 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 56.4

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.4

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

   9. Applied rewrites 67.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 75.7

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 75.7

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 75.7

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

   11. Applied rewrites 75.7%

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

   if 2e-140 < k < 1.32e9

   1. Initial program 44.3%

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

   3. Taylor expanded in 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. Applied rewrites 71.8%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 94.4%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 94.4

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

   12. Applied rewrites 94.4%

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

   if 1.32e9 < k

   1. Initial program 51.8%

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

   3. Taylor expanded in 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. Applied rewrites 79.4%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 80.8%

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

   10. Taylor expanded in k around inf

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

       1. associate-/l* N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-sin.f64 84.2

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

   12. Applied rewrites 84.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 1320000000:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\tan k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{elif}\;k \leq 1320000000:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(t\_m \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 2e-140)
    (* (/ l (* t_m k)) (/ l (* t_m (* t_m k))))
    (if (<= k 1320000000.0)
      (/
       2.0
       (* (* (/ (sin k) l) (fma k k (* 2.0 (* t_m t_m)))) (* t_m (/ k l))))
      (/ 2.0 (* (* (tan k) (/ (sin k) (* l l))) (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2e-140) {
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	} else if (k <= 1320000000.0) {
		tmp = 2.0 / (((sin(k) / l) * fma(k, k, (2.0 * (t_m * t_m)))) * (t_m * (k / l)));
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) / (l * l))) * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2e-140)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * Float64(t_m * k))));
	elseif (k <= 1320000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * fma(k, k, Float64(2.0 * Float64(t_m * t_m)))) * Float64(t_m * Float64(k / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) / Float64(l * l))) * Float64(t_m * Float64(k * 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[k, 2e-140], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1320000000.0], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2e-140

   1. Initial program 56.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 52.9

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

   5. Applied rewrites 52.9%

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

      1. 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 56.4

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.4

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

   9. Applied rewrites 67.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 75.7

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 75.7

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 75.7

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

   11. Applied rewrites 75.7%

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

   if 2e-140 < k < 1.32e9

   1. Initial program 44.3%

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

   3. Taylor expanded in 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. Applied rewrites 71.8%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 94.4%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 94.4

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

   12. Applied rewrites 94.4%

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

   if 1.32e9 < k

   1. Initial program 51.8%

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

   3. Taylor expanded in 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. Applied rewrites 79.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.9

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

   10. Applied rewrites 72.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-140}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}\\ \mathbf{elif}\;k \leq 1320000000:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(t\_m \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+76)
    (/ 2.0 (* (* (/ (sin k) l) (fma k k (* 2.0 (* t_m t_m)))) (* t_m (/ k l))))
    (* l (/ (/ l (* t_m k)) (* 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) {
	double tmp;
	if ((l * l) <= 5e+76) {
		tmp = 2.0 / (((sin(k) / l) * fma(k, k, (2.0 * (t_m * t_m)))) * (t_m * (k / l)));
	} else {
		tmp = l * ((l / (t_m * k)) / (t_m * (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 (Float64(l * l) <= 5e+76)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) / l) * fma(k, k, Float64(2.0 * Float64(t_m * t_m)))) * Float64(t_m * Float64(k / l))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * 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[N[(l * l), $MachinePrecision], 5e+76], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999991e76

   1. Initial program 59.7%

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

   3. Taylor expanded in 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. Applied rewrites 76.9%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

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

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   9. Applied rewrites 97.6%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 89.2

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

   12. Applied rewrites 89.2%

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

   if 4.99999999999999991e76 < (*.f64 l l)

   1. Initial program 47.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.6

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

   5. Applied rewrites 51.6%

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

      1. 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 53.6

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.1

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

   9. Applied rewrites 61.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/r* N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 67.3

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 67.3

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 67.3

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 67.3

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

   11. Applied rewrites 67.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k}{\ell} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot k}}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+76)
    (/ 2.0 (* t_m (* (fma k k (* 2.0 (* t_m t_m))) (* (/ (sin k) l) (/ k l)))))
    (* l (/ (/ l (* t_m k)) (* 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) {
	double tmp;
	if ((l * l) <= 5e+76) {
		tmp = 2.0 / (t_m * (fma(k, k, (2.0 * (t_m * t_m))) * ((sin(k) / l) * (k / l))));
	} else {
		tmp = l * ((l / (t_m * k)) / (t_m * (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 (Float64(l * l) <= 5e+76)
		tmp = Float64(2.0 / Float64(t_m * Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) * Float64(Float64(sin(k) / l) * Float64(k / l)))));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * k)) / Float64(t_m * 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[N[(l * l), $MachinePrecision], 5e+76], N[(2.0 / N[(t$95$m * N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999991e76

   1. Initial program 59.7%

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

   3. Taylor expanded in 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. Applied rewrites 76.9%

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(\frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)}} \]
   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. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 86.3

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

   10. Applied rewrites 86.3%

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

   if 4.99999999999999991e76 < (*.f64 l l)

   1. Initial program 47.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.6

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

   5. Applied rewrites 51.6%

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

      1. 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 53.6

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 60.3

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

   7. Applied rewrites 60.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.1

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

   9. Applied rewrites 61.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. lift-*.f64 N/A

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

       7. associate-/r* N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 67.3

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 67.3

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 67.3

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 67.3

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

   11. Applied rewrites 67.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{2}{t \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t \cdot k}}{t \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot \cos k\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-113)
    (/
     2.0
     (* t_m (* (fma k k (* 2.0 (* t_m t_m))) (/ (* k k) (* l (* l (cos k)))))))
    (* (/ l (* t_m k)) (/ l (* 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) {
	double tmp;
	if (t_m <= 9e-113) {
		tmp = 2.0 / (t_m * (fma(k, k, (2.0 * (t_m * t_m))) * ((k * k) / (l * (l * cos(k))))));
	} else {
		tmp = (l / (t_m * k)) * (l / (t_m * (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 <= 9e-113)
		tmp = Float64(2.0 / Float64(t_m * Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) * Float64(Float64(k * k) / Float64(l * Float64(l * cos(k)))))));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * 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, 9e-113], N[(2.0 / N[(t$95$m * N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.0000000000000002e-113

   1. Initial program 48.6%

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

   3. Taylor expanded in 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. Applied rewrites 71.5%

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

   6. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.5

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

   8. Applied rewrites 65.5%

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

   if 9.0000000000000002e-113 < t

   1. Initial program 63.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 55.8

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

   5. Applied rewrites 55.8%

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

      1. 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 58.4

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 64.3

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

   9. Applied rewrites 64.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 75.5

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 75.5

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 75.5

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

   11. Applied rewrites 75.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \left(\ell \cdot \cos k\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell}, \frac{0.16666666666666666}{\ell}, \frac{1}{\ell \cdot \ell}\right)\right)\right)}\\ \end{array} \end{array} \]

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 9e-54)
    (* (/ l (* t_m k)) (/ l (* t_m (* t_m k))))
    (/
     2.0
     (*
      t_m
      (*
       (fma k k (* 2.0 (* t_m t_m)))
       (*
        (* k k)
        (fma (/ (* k k) l) (/ 0.16666666666666666 l) (/ 1.0 (* l l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-54) {
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	} else {
		tmp = 2.0 / (t_m * (fma(k, k, (2.0 * (t_m * t_m))) * ((k * k) * fma(((k * k) / l), (0.16666666666666666 / l), (1.0 / (l * l))))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-54)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * Float64(t_m * k))));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) * Float64(Float64(k * k) * fma(Float64(Float64(k * k) / l), Float64(0.16666666666666666 / l), Float64(1.0 / Float64(l * l)))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-54], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(0.16666666666666666 / l), $MachinePrecision] + N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.9999999999999997e-54

   1. Initial program 54.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.5

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

   5. Applied rewrites 53.5%

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

      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 57.2

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 66.9

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

   7. Applied rewrites 66.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.5

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

   9. Applied rewrites 67.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 76.5

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 76.5

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 76.5

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

   11. Applied rewrites 76.5%

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

   if 8.9999999999999997e-54 < k

   1. Initial program 51.7%

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

   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. Applied rewrites 79.9%

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

   6. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 64.9%

      \[\leadsto \frac{2}{t \cdot \left(\color{blue}{\left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(k \cdot k, \frac{0.16666666666666666}{\ell \cdot \ell}, \frac{1}{\ell \cdot \ell}\right)\right)} \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 64.0

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

   10. Applied rewrites 64.0%

       \[\leadsto \frac{2}{t \cdot \left(\left(\left(k \cdot k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k \cdot k}{\ell}, \frac{0.16666666666666666}{\ell}, \frac{1}{\ell \cdot \ell}\right)}\right) \cdot \mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \left(\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{k \cdot k}{\ell}, \frac{0.16666666666666666}{\ell}, \frac{1}{\ell \cdot \ell}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.8e-113)
    (/ 2.0 (* t_m (* (fma k k (* 2.0 (* t_m t_m))) (/ (* k k) (* l l)))))
    (* (/ l (* t_m k)) (/ l (* 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) {
	double tmp;
	if (t_m <= 8.8e-113) {
		tmp = 2.0 / (t_m * (fma(k, k, (2.0 * (t_m * t_m))) * ((k * k) / (l * l))));
	} else {
		tmp = (l / (t_m * k)) * (l / (t_m * (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 <= 8.8e-113)
		tmp = Float64(2.0 / Float64(t_m * Float64(fma(k, k, Float64(2.0 * Float64(t_m * t_m))) * Float64(Float64(k * k) / Float64(l * l)))));
	else
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * 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, 8.8e-113], N[(2.0 / N[(t$95$m * N[(N[(k * k + N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.80000000000000016e-113

   1. Initial program 48.6%

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

   3. Taylor expanded in 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. Applied rewrites 71.5%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   if 8.80000000000000016e-113 < t

   1. Initial program 63.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 55.8

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

   5. Applied rewrites 55.8%

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

      1. 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 58.4

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 64.3

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

   9. Applied rewrites 64.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 75.5

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 75.5

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 75.5

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

   11. Applied rewrites 75.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{t \cdot \left(\mathsf{fma}\left(k, k, 2 \cdot \left(t \cdot t\right)\right) \cdot \frac{k \cdot k}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 190:\\ \;\;\;\;\frac{\ell}{t\_m \cdot k} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t\_m}}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 190.0)
    (* (/ l (* t_m k)) (/ l (* t_m (* t_m k))))
    (/ (/ (* l l) t_m) (* t_m (* t_m (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 190.0) {
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	} else {
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 190.0d0) then
        tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)))
    else
        tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 190.0) {
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	} else {
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 190.0:
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)))
	else:
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 190.0)
		tmp = Float64(Float64(l / Float64(t_m * k)) * Float64(l / Float64(t_m * Float64(t_m * k))));
	else
		tmp = Float64(Float64(Float64(l * l) / 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 <= 190.0)
		tmp = (l / (t_m * k)) * (l / (t_m * (t_m * k)));
	else
		tmp = ((l * l) / t_m) / (t_m * (t_m * (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 190.0], N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 190

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.2

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

   5. Applied rewrites 55.2%

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

      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 59.1

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 68.5

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

   9. Applied rewrites 68.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. times-frac N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 76.8

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 76.8

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

       20. lift-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 76.8

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

   11. Applied rewrites 76.8%

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

   if 190 < k

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

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

      1. 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 47.5

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

   5. Applied rewrites 47.5%

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

      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. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 59.7

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

   7. Applied rewrites 59.7%

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

Alternative 15: 67.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot \left(t\_m \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 8.2e-162)
    (* l (/ l (* t_m (* k (* t_m (* t_m k))))))
    (* (/ l t_m) (/ l (* t_m (* t_m (* k k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e-162) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8.2d-162) then
        tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
    else
        tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8.2e-162) {
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	} else {
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 8.2e-162:
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))))
	else:
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8.2e-162)
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(k * Float64(t_m * Float64(t_m * k))))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 8.2e-162)
		tmp = l * (l / (t_m * (k * (t_m * (t_m * k)))));
	else
		tmp = (l / t_m) * (l / (t_m * (t_m * (k * k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 8.2e-162], N[(l * N[(l / N[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $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 < 8.20000000000000039e-162

   1. Initial program 56.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.9

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

   5. Applied rewrites 51.9%

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

      1. 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 55.5

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

      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. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 66.3

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

   7. Applied rewrites 66.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.0

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

   9. Applied rewrites 67.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 71.1

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 71.1

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 71.1

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

   11. Applied rewrites 71.1%

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

   if 8.20000000000000039e-162 < k

   1. Initial program 49.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 54.4

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

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
   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. lower-*.f64 N/A

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

      11. lower-*.f64 65.1

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

   7. Applied rewrites 65.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-162}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.2% accurate, 12.5× speedup

Math

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

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 (* t_m (* k (* 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 / (t_m * (k * (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 / (t_m * (k * (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 / (t_m * (k * (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 / (t_m * (k * (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(t_m * Float64(k * 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 / (t_m * (k * (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[(t$95$m * N[(k * N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

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

3. 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 53.0

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

5. Applied rewrites 53.0%

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

   1. 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 55.9

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

   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. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 63.5

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

7. Applied rewrites 63.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   5. lower-*.f64 63.9

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

9. Applied rewrites 63.9%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

    3. *-commutative N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(\left(\left(k \cdot t\right) \cdot t\right) \cdot t\right)}} \cdot \ell \]

    4. associate-*r* N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right) \cdot t}} \cdot \ell \]

    6. lower-*.f64 67.7

       \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(\left(k \cdot t\right) \cdot t\right)\right)} \cdot t} \cdot \ell \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right) \cdot t} \cdot \ell \]

    8. *-commutative N/A

       \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(k \cdot t\right)\right)}\right) \cdot t} \cdot \ell \]

    9. lower-*.f64 67.7

       \[\leadsto \frac{\ell}{\left(k \cdot \color{blue}{\left(t \cdot \left(k \cdot t\right)\right)}\right) \cdot t} \cdot \ell \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \cdot t} \cdot \ell \]

    11. *-commutative N/A

        \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot k\right)}\right)\right) \cdot t} \cdot \ell \]

    12. lower-*.f64 67.7

        \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot \color{blue}{\left(t \cdot k\right)}\right)\right) \cdot t} \cdot \ell \]

11. Applied rewrites 67.7%

    \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right) \cdot t}} \cdot \ell \]

12. Final simplification 67.7%

    \[\leadsto \ell \cdot \frac{\ell}{t \cdot \left(k \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
13. Add Preprocessing

Alternative 17: 62.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\ell}{\left(t\_m \cdot k\right) \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ l (* (* t_m k) (* k (* t_m t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(l / Float64(Float64(t_m * k) * Float64(k * Float64(t_m * t_m))))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * (l / ((t_m * k) * (k * (t_m * t_m)))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(l / N[(N[(t$95$m * k), $MachinePrecision] * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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 53.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 53.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. 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 55.9

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   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. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot k\right)} \cdot \ell \]

   15. associate-*l* N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   17. lower-*.f64 63.5

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

7. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot k\right)\right)} \cdot \ell \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

   3. associate-*r* N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot k\right)}} \cdot \ell \]

   4. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(\left(t \cdot t\right) \cdot k\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

   7. *-commutative N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right)} \cdot \left(k \cdot t\right)} \cdot \ell \]

   9. lower-*.f64 64.3

      \[\leadsto \frac{\ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot t\right)}} \cdot \ell \]

9. Applied rewrites 64.3%

   \[\leadsto \frac{\ell}{\color{blue}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot t\right)}} \cdot \ell \]

10. Final simplification 64.3%

    \[\leadsto \ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)} \]
11. Add Preprocessing

Alternative 18: 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 53.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. 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 53.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

5. Applied rewrites 53.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \]
6. Step-by-step derivation

   1. 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 55.9

      \[\leadsto \color{blue}{\frac{\ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot k\right)}} \cdot \ell \]

   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. lift-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \left(\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot k\right)} \cdot \ell \]

   15. associate-*l* N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\ell}{k \cdot \color{blue}{\left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)}} \cdot \ell \]

   17. lower-*.f64 63.5

       \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot k\right)}\right)} \cdot \ell \]

7. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{\ell}{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot k\right)\right)} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(t \cdot \left(t \cdot k\right)\right)}\right)} \cdot \ell \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)} \cdot \ell \]

   3. *-commutative N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

   5. lower-*.f64 63.9

      \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \left(\color{blue}{\left(k \cdot t\right)} \cdot t\right)\right)} \cdot \ell \]

9. Applied rewrites 63.9%

   \[\leadsto \frac{\ell}{k \cdot \left(t \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot t\right)}\right)} \cdot \ell \]

10. Final simplification 63.9%

    \[\leadsto \ell \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot \left(t \cdot k\right)\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))