Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 90.2%

Time: 14.7s

Alternatives: 16

Speedup: 10.7×

• 26.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right) \cdot t\_m}{\ell}, k, \left(\frac{{t\_m}^{3}}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right) \cdot \frac{2}{\cos k}\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \frac{\left(\frac{\sin k}{\ell} \cdot {t\_m}^{3}\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e+25)
    (/
     2.0
     (fma
      (/ (* (* (/ (* (sin k) k) l) (tan k)) t_m) l)
      k
      (* (* (/ (pow t_m 3.0) l) (/ (pow (sin k) 2.0) l)) (/ 2.0 (cos k)))))
    (if (<= t_m 7.2e+106)
      (/
       2.0
       (*
        (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
        (/ (* (* (/ (sin k) l) (pow t_m 3.0)) (tan k)) l)))
      (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e+25) {
		tmp = 2.0 / fma((((((sin(k) * k) / l) * tan(k)) * t_m) / l), k, (((pow(t_m, 3.0) / l) * (pow(sin(k), 2.0) / l)) * (2.0 / cos(k))));
	} else if (t_m <= 7.2e+106) {
		tmp = 2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * ((((sin(k) / l) * pow(t_m, 3.0)) * tan(k)) / l));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e+25)
		tmp = Float64(2.0 / fma(Float64(Float64(Float64(Float64(Float64(sin(k) * k) / l) * tan(k)) * t_m) / l), k, Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64((sin(k) ^ 2.0) / l)) * Float64(2.0 / cos(k)))));
	elseif (t_m <= 7.2e+106)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(sin(k) / l) * (t_m ^ 3.0)) * tan(k)) / l)));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e+25], N[(2.0 / N[(N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * k + N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+106], N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.1999999999999997e25

   1. Initial program 42.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 81.0%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{\frac{t}{\ell} \cdot \left(\left(\tan k \cdot \sin k\right) \cdot k\right)}{\ell}, k, \frac{2}{\cos k} \cdot \left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 83.1%

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

   if 5.1999999999999997e25 < t < 7.2000000000000002e106

   1. Initial program 73.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   if 7.2000000000000002e106 < t

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 58.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 58.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 58.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   9. Applied rewrites 52.6%

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

   11. Applied rewrites 60.4%

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

   13. Applied rewrites 82.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{\left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right) \cdot t}{\ell}, k, \left(\frac{{t}^{3}}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right) \cdot \frac{2}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \frac{\left(\frac{\sin k}{\ell} \cdot {t}^{3}\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right) + 1\right) \cdot \frac{\left(\frac{\sin k}{\ell} \cdot {t\_m}^{3}\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-13)
    (/ 2.0 (* (* (/ t_m l) k) (* (/ (* (sin k) k) l) (tan k))))
    (if (<= t_m 7.2e+106)
      (/
       2.0
       (*
        (+ (+ (pow (/ k t_m) 2.0) 1.0) 1.0)
        (/ (* (* (/ (sin k) l) (pow t_m 3.0)) (tan k)) l)))
      (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-13) {
		tmp = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	} else if (t_m <= 7.2e+106) {
		tmp = 2.0 / (((pow((k / t_m), 2.0) + 1.0) + 1.0) * ((((sin(k) / l) * pow(t_m, 3.0)) * tan(k)) / l));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.5d-13) then
        tmp = 2.0d0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)))
    else if (t_m <= 7.2d+106) then
        tmp = 2.0d0 / (((((k / t_m) ** 2.0d0) + 1.0d0) + 1.0d0) * ((((sin(k) / l) * (t_m ** 3.0d0)) * tan(k)) / l))
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-13) {
		tmp = 2.0 / (((t_m / l) * k) * (((Math.sin(k) * k) / l) * Math.tan(k)));
	} else if (t_m <= 7.2e+106) {
		tmp = 2.0 / (((Math.pow((k / t_m), 2.0) + 1.0) + 1.0) * ((((Math.sin(k) / l) * Math.pow(t_m, 3.0)) * Math.tan(k)) / l));
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.5e-13:
		tmp = 2.0 / (((t_m / l) * k) * (((math.sin(k) * k) / l) * math.tan(k)))
	elif t_m <= 7.2e+106:
		tmp = 2.0 / (((math.pow((k / t_m), 2.0) + 1.0) + 1.0) * ((((math.sin(k) / l) * math.pow(t_m, 3.0)) * math.tan(k)) / l))
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.5e-13)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(Float64(sin(k) * k) / l) * tan(k))));
	elseif (t_m <= 7.2e+106)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 1.0) + 1.0) * Float64(Float64(Float64(Float64(sin(k) / l) * (t_m ^ 3.0)) * tan(k)) / l)));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.5e-13)
		tmp = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	elseif (t_m <= 7.2e+106)
		tmp = 2.0 / (((((k / t_m) ^ 2.0) + 1.0) + 1.0) * ((((sin(k) / l) * (t_m ^ 3.0)) * tan(k)) / l));
	else
		tmp = (l / ((k * t_m) ^ 2.0)) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.5e-13], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+106], N[(2.0 / N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.49999999999999979e-13

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.2

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}{{t}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 65.7%

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

   9. Applied rewrites 64.0%

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

   11. Applied rewrites 78.1%

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

   if 5.49999999999999979e-13 < t < 7.2000000000000002e106

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

   if 7.2000000000000002e106 < t

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 58.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 58.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 58.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   9. Applied rewrites 52.6%

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

   11. Applied rewrites 60.4%

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

   13. Applied rewrites 82.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right) \cdot \frac{\left(\frac{\sin k}{\ell} \cdot {t}^{3}\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\_m\right)\right) \cdot \frac{t\_m \cdot t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-13)
    (/ 2.0 (* (* (/ t_m l) k) (* (/ (* (sin k) k) l) (tan k))))
    (if (<= t_m 6.2e+127)
      (/
       2.0
       (*
        (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (* (/ (sin k) l) t_m))
        (/ (* t_m t_m) l)))
      (* (/ l t_m) (/ l (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 tmp;
	if (t_m <= 2.6e-13) {
		tmp = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	} else if (t_m <= 6.2e+127) {
		tmp = 2.0 / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * ((sin(k) / l) * t_m)) * ((t_m * t_m) / l));
	} else {
		tmp = (l / t_m) * (l / pow((k * t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.6d-13) then
        tmp = 2.0d0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)))
    else if (t_m <= 6.2d+127) then
        tmp = 2.0d0 / ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * ((sin(k) / l) * t_m)) * ((t_m * t_m) / l))
    else
        tmp = (l / t_m) * (l / ((k * t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-13) {
		tmp = 2.0 / (((t_m / l) * k) * (((Math.sin(k) * k) / l) * Math.tan(k)));
	} else if (t_m <= 6.2e+127) {
		tmp = 2.0 / ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * ((Math.sin(k) / l) * t_m)) * ((t_m * t_m) / l));
	} else {
		tmp = (l / t_m) * (l / Math.pow((k * t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.6e-13:
		tmp = 2.0 / (((t_m / l) * k) * (((math.sin(k) * k) / l) * math.tan(k)))
	elif t_m <= 6.2e+127:
		tmp = 2.0 / ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * ((math.sin(k) / l) * t_m)) * ((t_m * t_m) / l))
	else:
		tmp = (l / t_m) * (l / math.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)
	tmp = 0.0
	if (t_m <= 2.6e-13)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(Float64(sin(k) * k) / l) * tan(k))));
	elseif (t_m <= 6.2e+127)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(Float64(sin(k) / l) * t_m)) * Float64(Float64(t_m * t_m) / l)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / (Float64(k * t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.6e-13)
		tmp = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	elseif (t_m <= 6.2e+127)
		tmp = 2.0 / ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * ((sin(k) / l) * t_m)) * ((t_m * t_m) / l));
	else
		tmp = (l / t_m) * (l / ((k * t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-13], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+127], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.6e-13

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.2

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}{{t}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 65.7%

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

   9. Applied rewrites 64.0%

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

   11. Applied rewrites 78.1%

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

   if 2.6e-13 < t < 6.2000000000000005e127

   1. Initial program 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 88.2%

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

   if 6.2000000000000005e127 < t

   1. Initial program 61.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 61.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 61.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 61.7

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 61.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.7

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

   7. Applied rewrites 54.7%

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

   9. Applied rewrites 54.7%

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

   11. Applied rewrites 61.3%

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

   13. Applied rewrites 86.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right) \cdot \frac{t \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{{\left(k \cdot t\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\left(\frac{\sin k}{\ell} \cdot {t\_m}^{3}\right) \cdot \tan k}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (* (* (/ t_m l) k) (* (/ (* (sin k) k) l) (tan k))))))
   (*
    t_s
    (if (<= t_m 1.8e-11)
      t_2
      (if (<= t_m 5.8e+60)
        (/ 2.0 (* 2.0 (/ (* (* (/ (sin k) l) (pow t_m 3.0)) (tan k)) l)))
        (if (<= t_m 1.3e+82) t_2 (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	double tmp;
	if (t_m <= 1.8e-11) {
		tmp = t_2;
	} else if (t_m <= 5.8e+60) {
		tmp = 2.0 / (2.0 * ((((sin(k) / l) * pow(t_m, 3.0)) * tan(k)) / l));
	} else if (t_m <= 1.3e+82) {
		tmp = t_2;
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)))
    if (t_m <= 1.8d-11) then
        tmp = t_2
    else if (t_m <= 5.8d+60) then
        tmp = 2.0d0 / (2.0d0 * ((((sin(k) / l) * (t_m ** 3.0d0)) * tan(k)) / l))
    else if (t_m <= 1.3d+82) then
        tmp = t_2
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (((t_m / l) * k) * (((Math.sin(k) * k) / l) * Math.tan(k)));
	double tmp;
	if (t_m <= 1.8e-11) {
		tmp = t_2;
	} else if (t_m <= 5.8e+60) {
		tmp = 2.0 / (2.0 * ((((Math.sin(k) / l) * Math.pow(t_m, 3.0)) * Math.tan(k)) / l));
	} else if (t_m <= 1.3e+82) {
		tmp = t_2;
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 / (((t_m / l) * k) * (((math.sin(k) * k) / l) * math.tan(k)))
	tmp = 0
	if t_m <= 1.8e-11:
		tmp = t_2
	elif t_m <= 5.8e+60:
		tmp = 2.0 / (2.0 * ((((math.sin(k) / l) * math.pow(t_m, 3.0)) * math.tan(k)) / l))
	elif t_m <= 1.3e+82:
		tmp = t_2
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(Float64(sin(k) * k) / l) * tan(k))))
	tmp = 0.0
	if (t_m <= 1.8e-11)
		tmp = t_2;
	elseif (t_m <= 5.8e+60)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(sin(k) / l) * (t_m ^ 3.0)) * tan(k)) / l)));
	elseif (t_m <= 1.3e+82)
		tmp = t_2;
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	tmp = 0.0;
	if (t_m <= 1.8e-11)
		tmp = t_2;
	elseif (t_m <= 5.8e+60)
		tmp = 2.0 / (2.0 * ((((sin(k) / l) * (t_m ^ 3.0)) * tan(k)) / l));
	elseif (t_m <= 1.3e+82)
		tmp = t_2;
	else
		tmp = (l / ((k * t_m) ^ 2.0)) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.8e-11], t$95$2, If[LessEqual[t$95$m, 5.8e+60], N[(2.0 / N[(2.0 * N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+82], t$95$2, N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.79999999999999992e-11 or 5.79999999999999999e60 < t < 1.2999999999999999e82

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.5

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

   4. Applied rewrites 10.5%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}{{t}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 65.0%

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

   9. Applied rewrites 63.3%

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

   11. Applied rewrites 78.7%

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

   if 1.79999999999999992e-11 < t < 5.79999999999999999e60

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      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. associate-/r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 94.4

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 94.4%

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

   if 1.2999999999999999e82 < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 60.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 60.0

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 59.8%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   9. Applied rewrites 54.3%

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

   11. Applied rewrites 61.5%

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

   13. Applied rewrites 83.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\left(\frac{\sin k}{\ell} \cdot {t}^{3}\right) \cdot \tan k}{\ell}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{{t\_m}^{-3}}{\frac{k}{\ell}} \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (* (* (/ t_m l) k) (* (/ (* (sin k) k) l) (tan k))))))
   (*
    t_s
    (if (<= t_m 4.5e-12)
      t_2
      (if (<= t_m 5.8e+60)
        (* (/ (pow t_m -3.0) (/ k l)) (/ l k))
        (if (<= t_m 1.3e+82) t_2 (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	double tmp;
	if (t_m <= 4.5e-12) {
		tmp = t_2;
	} else if (t_m <= 5.8e+60) {
		tmp = (pow(t_m, -3.0) / (k / l)) * (l / k);
	} else if (t_m <= 1.3e+82) {
		tmp = t_2;
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 2.0d0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)))
    if (t_m <= 4.5d-12) then
        tmp = t_2
    else if (t_m <= 5.8d+60) then
        tmp = ((t_m ** (-3.0d0)) / (k / l)) * (l / k)
    else if (t_m <= 1.3d+82) then
        tmp = t_2
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / (((t_m / l) * k) * (((Math.sin(k) * k) / l) * Math.tan(k)));
	double tmp;
	if (t_m <= 4.5e-12) {
		tmp = t_2;
	} else if (t_m <= 5.8e+60) {
		tmp = (Math.pow(t_m, -3.0) / (k / l)) * (l / k);
	} else if (t_m <= 1.3e+82) {
		tmp = t_2;
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 2.0 / (((t_m / l) * k) * (((math.sin(k) * k) / l) * math.tan(k)))
	tmp = 0
	if t_m <= 4.5e-12:
		tmp = t_2
	elif t_m <= 5.8e+60:
		tmp = (math.pow(t_m, -3.0) / (k / l)) * (l / k)
	elif t_m <= 1.3e+82:
		tmp = t_2
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(Float64(sin(k) * k) / l) * tan(k))))
	tmp = 0.0
	if (t_m <= 4.5e-12)
		tmp = t_2;
	elseif (t_m <= 5.8e+60)
		tmp = Float64(Float64((t_m ^ -3.0) / Float64(k / l)) * Float64(l / k));
	elseif (t_m <= 1.3e+82)
		tmp = t_2;
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	tmp = 0.0;
	if (t_m <= 4.5e-12)
		tmp = t_2;
	elseif (t_m <= 5.8e+60)
		tmp = ((t_m ^ -3.0) / (k / l)) * (l / k);
	elseif (t_m <= 1.3e+82)
		tmp = t_2;
	else
		tmp = (l / ((k * t_m) ^ 2.0)) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.5e-12], t$95$2, If[LessEqual[t$95$m, 5.8e+60], N[(N[(N[Power[t$95$m, -3.0], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+82], t$95$2, N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.49999999999999981e-12 or 5.79999999999999999e60 < t < 1.2999999999999999e82

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.5

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

   4. Applied rewrites 10.5%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}{{t}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 65.0%

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

   9. Applied rewrites 63.3%

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

   11. Applied rewrites 78.7%

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

   if 4.49999999999999981e-12 < t < 5.79999999999999999e60

   1. Initial program 78.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 78.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.7

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 89.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   9. Applied rewrites 56.8%

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

   11. Applied rewrites 88.9%

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

   if 1.2999999999999999e82 < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 60.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 60.0

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 59.8%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

   9. Applied rewrites 54.3%

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

   11. Applied rewrites 61.5%

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

   13. Applied rewrites 83.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{{t}^{-3}}{\frac{k}{\ell}} \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \tan k\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-12)
    (/ 2.0 (* (* (/ t_m l) k) (* (/ (* (sin k) k) l) (tan k))))
    (/ 2.0 (* (* (* (* (* (/ t_m l) t_m) (tan k)) (sin k)) (/ t_m l)) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-12) {
		tmp = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	} else {
		tmp = 2.0 / ((((((t_m / l) * t_m) * tan(k)) * sin(k)) * (t_m / l)) * 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.2d-12) then
        tmp = 2.0d0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)))
    else
        tmp = 2.0d0 / ((((((t_m / l) * t_m) * tan(k)) * sin(k)) * (t_m / l)) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-12) {
		tmp = 2.0 / (((t_m / l) * k) * (((Math.sin(k) * k) / l) * Math.tan(k)));
	} else {
		tmp = 2.0 / ((((((t_m / l) * t_m) * Math.tan(k)) * Math.sin(k)) * (t_m / l)) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.2e-12:
		tmp = 2.0 / (((t_m / l) * k) * (((math.sin(k) * k) / l) * math.tan(k)))
	else:
		tmp = 2.0 / ((((((t_m / l) * t_m) * math.tan(k)) * math.sin(k)) * (t_m / l)) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-12)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(Float64(sin(k) * k) / l) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * tan(k)) * sin(k)) * Float64(t_m / l)) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.2e-12)
		tmp = 2.0 / (((t_m / l) * k) * (((sin(k) * k) / l) * tan(k)));
	else
		tmp = 2.0 / ((((((t_m / l) * t_m) * tan(k)) * sin(k)) * (t_m / l)) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-12], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.19999999999999965e-12

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.2

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

   4. Applied rewrites 10.2%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}{{t}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 65.7%

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

   9. Applied rewrites 64.0%

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

   11. Applied rewrites 78.1%

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

   if 5.19999999999999965e-12 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

   7. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-tan.f64 81.5

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

   9. Applied rewrites 81.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{\sin k \cdot k}{\ell} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \tan k\right) \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell \cdot \ell} \cdot k\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot k\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{{t\_m}^{-3}}{\frac{k}{\ell}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8.5e-36)
    (/ 2.0 (* (* (/ t_m (* l l)) k) (* (* (sin k) (tan k)) k)))
    (if (<= t_m 7.2e+106)
      (* (/ (pow t_m -3.0) (/ k l)) (/ l k))
      (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-36) {
		tmp = 2.0 / (((t_m / (l * l)) * k) * ((sin(k) * tan(k)) * k));
	} else if (t_m <= 7.2e+106) {
		tmp = (pow(t_m, -3.0) / (k / l)) * (l / k);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 8.5d-36) then
        tmp = 2.0d0 / (((t_m / (l * l)) * k) * ((sin(k) * tan(k)) * k))
    else if (t_m <= 7.2d+106) then
        tmp = ((t_m ** (-3.0d0)) / (k / l)) * (l / k)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8.5e-36) {
		tmp = 2.0 / (((t_m / (l * l)) * k) * ((Math.sin(k) * Math.tan(k)) * k));
	} else if (t_m <= 7.2e+106) {
		tmp = (Math.pow(t_m, -3.0) / (k / l)) * (l / k);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 8.5e-36:
		tmp = 2.0 / (((t_m / (l * l)) * k) * ((math.sin(k) * math.tan(k)) * k))
	elif t_m <= 7.2e+106:
		tmp = (math.pow(t_m, -3.0) / (k / l)) * (l / k)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8.5e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / Float64(l * l)) * k) * Float64(Float64(sin(k) * tan(k)) * k)));
	elseif (t_m <= 7.2e+106)
		tmp = Float64(Float64((t_m ^ -3.0) / Float64(k / l)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8.5e-36)
		tmp = 2.0 / (((t_m / (l * l)) * k) * ((sin(k) * tan(k)) * k));
	elseif (t_m <= 7.2e+106)
		tmp = ((t_m ^ -3.0) / (k / l)) * (l / k);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8.5e-36], N[(2.0 / N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+106], N[(N[(N[Power[t$95$m, -3.0], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.5000000000000007e-36

   1. Initial program 42.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 10.0

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{e^{3 \cdot \log t - 2 \cdot \log \ell} \cdot {\sin k}^{2}}{{t}^{2} \cdot \cos k}}} \]

      2. times-frac N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 66.4%

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

   9. Applied rewrites 68.8%

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

   if 8.5000000000000007e-36 < t < 7.2000000000000002e106

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 66.4

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 66.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.4

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 72.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

   9. Applied rewrites 51.6%

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

   11. Applied rewrites 73.0%

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

   if 7.2000000000000002e106 < t

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 58.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 58.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 58.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   9. Applied rewrites 52.6%

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

   11. Applied rewrites 60.4%

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

   13. Applied rewrites 82.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell \cdot \ell} \cdot k\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot k\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{{t}^{-3}}{\frac{k}{\ell}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(\sin k \cdot k\right) \cdot k\right) \cdot \left(\tan k \cdot t\_m\right)}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{{t\_m}^{-3}}{\frac{k}{\ell}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-26)
    (/ (* (* l l) 2.0) (* (* (* (sin k) k) k) (* (tan k) t_m)))
    (if (<= t_m 7.2e+106)
      (* (/ (pow t_m -3.0) (/ k l)) (/ l k))
      (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-26) {
		tmp = ((l * l) * 2.0) / (((sin(k) * k) * k) * (tan(k) * t_m));
	} else if (t_m <= 7.2e+106) {
		tmp = (pow(t_m, -3.0) / (k / l)) * (l / k);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-26) then
        tmp = ((l * l) * 2.0d0) / (((sin(k) * k) * k) * (tan(k) * t_m))
    else if (t_m <= 7.2d+106) then
        tmp = ((t_m ** (-3.0d0)) / (k / l)) * (l / k)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-26) {
		tmp = ((l * l) * 2.0) / (((Math.sin(k) * k) * k) * (Math.tan(k) * t_m));
	} else if (t_m <= 7.2e+106) {
		tmp = (Math.pow(t_m, -3.0) / (k / l)) * (l / k);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-26:
		tmp = ((l * l) * 2.0) / (((math.sin(k) * k) * k) * (math.tan(k) * t_m))
	elif t_m <= 7.2e+106:
		tmp = (math.pow(t_m, -3.0) / (k / l)) * (l / k)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-26)
		tmp = Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(Float64(sin(k) * k) * k) * Float64(tan(k) * t_m)));
	elseif (t_m <= 7.2e+106)
		tmp = Float64(Float64((t_m ^ -3.0) / Float64(k / l)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-26)
		tmp = ((l * l) * 2.0) / (((sin(k) * k) * k) * (tan(k) * t_m));
	elseif (t_m <= 7.2e+106)
		tmp = ((t_m ^ -3.0) / (k / l)) * (l / k);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-26], N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e+106], N[(N[(N[Power[t$95$m, -3.0], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.30000000000000009e-26

   1. Initial program 42.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 42.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 42.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 42.0

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 43.5%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.1

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

   7. Applied rewrites 50.1%

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

   9. Applied rewrites 50.1%

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

   10. Taylor expanded in t around 0

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-sin.f64 64.7

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

   12. Applied rewrites 64.7%

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

   14. Applied rewrites 64.7%

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

   if 2.30000000000000009e-26 < t < 7.2000000000000002e106

   1. Initial program 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 68.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 68.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 74.8%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 53.1

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

   7. Applied rewrites 53.1%

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

   9. Applied rewrites 53.1%

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

   11. Applied rewrites 75.1%

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

   if 7.2000000000000002e106 < t

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 58.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 58.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 58.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 58.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.6

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

   7. Applied rewrites 52.6%

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

   9. Applied rewrites 52.6%

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

   11. Applied rewrites 60.4%

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

   13. Applied rewrites 82.6%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(\sin k \cdot k\right) \cdot k\right) \cdot \left(\tan k \cdot t\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{{t}^{-3}}{\frac{k}{\ell}} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}}{\frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e+55)
    (/ (* (/ (/ (/ l k) k) t_m) (/ l t_m)) t_m)
    (/ (/ l (pow (* k t_m) 2.0)) (/ t_m l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e+55) {
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m;
	} else {
		tmp = (l / pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d+55) then
        tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) / (t_m / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e+55) {
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m;
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) / (t_m / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e+55:
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) / (t_m / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e+55)
		tmp = Float64(Float64(Float64(Float64(Float64(l / k) / k) / t_m) * Float64(l / t_m)) / t_m);
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) / Float64(t_m / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e+55)
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m;
	else
		tmp = (l / ((k * t_m) ^ 2.0)) / (t_m / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e+55], N[(N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6999999999999999e55

   1. Initial program 44.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 44.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 44.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 44.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 46.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.4

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

   7. Applied rewrites 50.4%

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

   9. Applied rewrites 50.4%

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

   11. Applied rewrites 55.1%

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

   13. Applied rewrites 67.7%

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

   if 1.6999999999999999e55 < t

   1. Initial program 60.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 60.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 60.1

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 61.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.4

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

   7. Applied rewrites 52.4%

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

   9. Applied rewrites 52.4%

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

   11. Applied rewrites 59.8%

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

   13. Applied rewrites 78.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t} \cdot \frac{\ell}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{{\left(k \cdot t\right)}^{2}}}{\frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{{\left(k \cdot t\_m\right)}^{2}}\\ \end{array} \end{array} \]

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 1.7e+55)
    (/ (* (/ (/ (/ l k) k) t_m) (/ l t_m)) t_m)
    (* (/ l t_m) (/ l (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 tmp;
	if (t_m <= 1.7e+55) {
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m;
	} else {
		tmp = (l / t_m) * (l / pow((k * t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d+55) then
        tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m
    else
        tmp = (l / t_m) * (l / ((k * t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e+55) {
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m;
	} else {
		tmp = (l / t_m) * (l / Math.pow((k * t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e+55:
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m
	else:
		tmp = (l / t_m) * (l / math.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)
	tmp = 0.0
	if (t_m <= 1.7e+55)
		tmp = Float64(Float64(Float64(Float64(Float64(l / k) / k) / t_m) * Float64(l / t_m)) / t_m);
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / (Float64(k * t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e+55)
		tmp = ((((l / k) / k) / t_m) * (l / t_m)) / t_m;
	else
		tmp = (l / t_m) * (l / ((k * t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e+55], N[(N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.6999999999999999e55

   1. Initial program 44.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 44.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 44.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 44.6

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 46.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 50.4

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

   7. Applied rewrites 50.4%

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

   9. Applied rewrites 50.4%

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

   11. Applied rewrites 55.1%

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

   13. Applied rewrites 67.7%

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

   if 1.6999999999999999e55 < t

   1. Initial program 60.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 60.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 60.1

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 61.4%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.4

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

   7. Applied rewrites 52.4%

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

   9. Applied rewrites 52.4%

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

   11. Applied rewrites 59.8%

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

   13. Applied rewrites 78.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t} \cdot \frac{\ell}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{{\left(k \cdot t\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot 2}\\ \end{array} \end{array} \]

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 3.5e-151)
    (/ (* (/ l k) l) (* (* (* t_m t_m) k) t_m))
    (/ 2.0 (* (* (/ (* (* (* k k) t_m) t_m) l) (/ t_m l)) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e-151) {
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * (t_m / l)) * 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.5d-151) then
        tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m)
    else
        tmp = 2.0d0 / ((((((k * k) * t_m) * t_m) / l) * (t_m / l)) * 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e-151) {
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * (t_m / l)) * 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.5e-151:
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m)
	else:
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * (t_m / l)) * 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.5e-151)
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(Float64(t_m * t_m) * k) * t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) / l) * Float64(t_m / l)) * 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.5e-151)
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m);
	else
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * (t_m / l)) * 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.5e-151], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.49999999999999995e-151

   1. Initial program 46.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 46.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 46.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 46.7

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

      15. lift-*.f64 N/A

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

   3. Applied rewrites 49.5%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 48.6

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

   7. Applied rewrites 48.6%

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

   9. Applied rewrites 58.7%

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

   11. Applied rewrites 63.8%

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

   if 3.49999999999999995e-151 < k

   1. Initial program 50.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

   7. Applied rewrites 56.6%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}\right) \cdot 2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}\right) \cdot 2} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2} \cdot \color{blue}{\left(t \cdot t\right)}}{\ell}\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot t}}{\ell}\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot t}}{\ell}\right) \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\left({k}^{2} \cdot t\right)} \cdot t}{\ell}\right) \cdot 2} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t}{\ell}\right) \cdot 2} \]

      7. lower-*.f64 65.6

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot t}{\ell}\right) \cdot 2} \]

   10. Applied rewrites 65.6%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell}}\right) \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\frac{\frac{\ell}{k}}{k}}{t\_m} \cdot \frac{\ell}{t\_m}}{t\_m} \end{array} \]

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 k) k) t_m) (/ l 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 / k) / k) / t_m) * (l / 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 / k) / k) / t_m) * (l / 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 / k) / k) / t_m) * (l / 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 / k) / k) / t_m) * (l / 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(Float64(Float64(Float64(Float64(l / k) / k) / t_m) * Float64(l / 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 / k) / k) / t_m) * (l / 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[(N[(N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   5. lower-/.f64 48.4

      \[\leadsto \frac{\color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   9. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. lower-+.f64 48.4

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   14. lower-*.f64 48.4

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

3. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 50.9

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

7. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 50.9%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
10. Step-by-step derivation

11. Applied rewrites 56.2%

    \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
12. Step-by-step derivation

13. Applied rewrites 67.1%

    \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}}{\color{blue}{t}} \]

14. Final simplification 67.1%

    \[\leadsto \frac{\frac{\frac{\frac{\ell}{k}}{k}}{t} \cdot \frac{\ell}{t}}{t} \]
15. Add Preprocessing

Alternative 13: 65.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 3.5e-151)
    (/ (* (/ l k) l) (* (* (* t_m t_m) k) t_m))
    (/ (* (/ l t_m) l) (* (* (* k k) t_m) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e-151) {
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m);
	} else {
		tmp = ((l / t_m) * l) / (((k * k) * t_m) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.5d-151) then
        tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m)
    else
        tmp = ((l / t_m) * l) / (((k * k) * t_m) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.5e-151) {
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m);
	} else {
		tmp = ((l / t_m) * l) / (((k * k) * t_m) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.5e-151:
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m)
	else:
		tmp = ((l / t_m) * l) / (((k * k) * t_m) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.5e-151)
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(Float64(t_m * t_m) * k) * t_m));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(Float64(Float64(k * k) * t_m) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.5e-151)
		tmp = ((l / k) * l) / (((t_m * t_m) * k) * t_m);
	else
		tmp = ((l / t_m) * l) / (((k * k) * t_m) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.5e-151], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.49999999999999995e-151

   1. Initial program 46.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      5. lower-/.f64 46.7

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      9. associate-+l+ N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. lower-+.f64 46.7

          \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      14. lower-*.f64 46.7

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   3. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 48.6

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   7. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.7%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot {t}^{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.8%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

   if 3.49999999999999995e-151 < k

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      5. lower-/.f64 50.3

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      9. associate-+l+ N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      11. lower-+.f64 50.3

          \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      14. lower-*.f64 50.3

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   3. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 53.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   7. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.7%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.3%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{k} \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \end{array} \]

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 k) l) (* (* t_m t_m) (* k 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 / k) * l) / ((t_m * t_m) * (k * 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 / k) * l) / ((t_m * t_m) * (k * 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 / k) * l) / ((t_m * t_m) * (k * 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 / k) * l) / ((t_m * t_m) * (k * 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(Float64(Float64(l / k) * l) / Float64(Float64(t_m * t_m) * Float64(k * 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 / k) * l) / ((t_m * t_m) * (k * 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[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   5. lower-/.f64 48.4

      \[\leadsto \frac{\color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   9. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. lower-+.f64 48.4

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   14. lower-*.f64 48.4

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

3. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 50.9

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

7. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 55.5%

   \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot {t}^{3}}} \]
10. Step-by-step derivation

11. Applied rewrites 60.1%

    \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]
12. Add Preprocessing

Alternative 15: 62.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot t\_m} \end{array} \]

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 k) l) (* (* (* t_m t_m) k) 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 / k) * l) / (((t_m * t_m) * k) * 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 / k) * l) / (((t_m * t_m) * k) * 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 / k) * l) / (((t_m * t_m) * k) * 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 / k) * l) / (((t_m * t_m) * k) * 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(Float64(Float64(l / k) * l) / Float64(Float64(Float64(t_m * t_m) * k) * 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 / k) * l) / (((t_m * t_m) * k) * 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[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   5. lower-/.f64 48.4

      \[\leadsto \frac{\color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   9. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. lower-+.f64 48.4

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   14. lower-*.f64 48.4

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

3. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 50.9

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

7. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 55.5%

   \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{k \cdot {t}^{3}}} \]
10. Step-by-step derivation

11. Applied rewrites 60.1%

    \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

12. Final simplification 60.1%

    \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot t} \]
13. Add Preprocessing

Alternative 16: 53.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l l) (* (* (* k k) t_m) (* t_m t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(Float64(k * k) * t_m) * Float64(t_m * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / (((k * k) * t_m) * (t_m * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   3. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   5. lower-/.f64 48.4

      \[\leadsto \frac{\color{blue}{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   9. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   11. lower-+.f64 48.4

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 2}}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   14. lower-*.f64 48.4

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

3. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k \cdot \left(\frac{\frac{\sin k}{\ell}}{\ell} \cdot {t}^{3}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 50.9

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

7. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 50.9%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
10. Step-by-step derivation

11. Applied rewrites 56.2%

    \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
12. Step-by-step derivation

13. Applied rewrites 53.4%

    \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024272 
(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))))