Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 92.7%

Time: 14.1s

Alternatives: 28

Speedup: 10.7×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-215}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\frac{\mathsf{fma}\left(\tan k, \left(k \cdot \sqrt{t\_m}\right) \cdot \frac{k}{\ell}, {t\_m}^{2.5} \cdot \frac{\tan k \cdot 2}{\ell}\right)}{t\_m} \cdot {t\_m}^{1.5}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell}}{\frac{\frac{\ell}{t\_m}}{\left(t\_m \cdot \sin k\right) \cdot \tan k}} \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-215)
    (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
    (if (<= t_m 2.5e+124)
      (*
       (/
        (/ l (sin k))
        (*
         (/
          (fma
           (tan k)
           (* (* k (sqrt t_m)) (/ k l))
           (* (pow t_m 2.5) (/ (* (tan k) 2.0) l)))
          t_m)
         (pow t_m 1.5)))
       2.0)
      (/
       2.0
       (*
        (/ (/ t_m l) (/ (/ l t_m) (* (* t_m (sin k)) (tan k))))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))))))

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 <= 1.55e-215) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 2.5e+124) {
		tmp = ((l / sin(k)) / ((fma(tan(k), ((k * sqrt(t_m)) * (k / l)), (pow(t_m, 2.5) * ((tan(k) * 2.0) / l))) / t_m) * pow(t_m, 1.5))) * 2.0;
	} else {
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	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 <= 1.55e-215)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 2.5e+124)
		tmp = Float64(Float64(Float64(l / sin(k)) / Float64(Float64(fma(tan(k), Float64(Float64(k * sqrt(t_m)) * Float64(k / l)), Float64((t_m ^ 2.5) * Float64(Float64(tan(k) * 2.0) / l))) / t_m) * (t_m ^ 1.5))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) / Float64(Float64(l / t_m) / Float64(Float64(t_m * sin(k)) * tan(k)))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.55e-215], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+124], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$m, 2.5], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.54999999999999997e-215

   1. Initial program 47.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 69.1%

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

   if 1.54999999999999997e-215 < t < 2.4999999999999998e124

   1. Initial program 56.8%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

   6. Applied rewrites 84.1%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{\sin k}{\ell} \cdot {t}^{\frac{3}{2}}\right) \cdot \color{blue}{\frac{2 \cdot \left(\frac{\sin k}{\ell \cdot \cos k} \cdot \sqrt{{t}^{5}}\right) + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k} \cdot \sqrt{t}}{t}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\sin k}{\ell} \cdot {t}^{\frac{3}{2}}\right) \cdot \color{blue}{\frac{2 \cdot \left(\frac{\sin k}{\ell \cdot \cos k} \cdot \sqrt{{t}^{5}}\right) + \frac{{k}^{2} \cdot \sin k}{\ell \cdot \cos k} \cdot \sqrt{t}}{t}}} \]

   9. Applied rewrites 84.7%

      \[\leadsto \frac{2}{\left(\frac{\sin k}{\ell} \cdot {t}^{1.5}\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(\sqrt{t} \cdot \left(k \cdot \frac{k}{\ell}\right), \frac{\sin k}{\cos k}, \frac{2 \cdot \frac{\sin k}{\ell}}{\cos k} \cdot \sqrt{{t}^{5}}\right)}{t}}} \]

   11. Applied rewrites 95.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k}}{\frac{\mathsf{fma}\left(\tan k, \left(k \cdot \sqrt{t}\right) \cdot \frac{k}{\ell}, {t}^{2.5} \cdot \frac{\tan k \cdot 2}{\ell}\right)}{t} \cdot {t}^{1.5}} \cdot 2} \]

   if 2.4999999999999998e124 < t

   1. Initial program 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 89.9

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

   6. Applied rewrites 89.9%

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

Alternative 2: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{\frac{\left({t\_m}^{1.5} \cdot \frac{\sin k}{\ell}\right) \cdot \left(\left(\left(t\_2 + 2\right) \cdot \tan k\right) \cdot {t\_m}^{1.5}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell}}{\frac{\frac{\ell}{t\_m}}{\left(t\_m \cdot \sin k\right) \cdot \tan k}} \cdot \left(\left(1 + t\_2\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 6.8e-29)
      (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
      (if (<= t_m 4.8e+143)
        (/
         2.0
         (/
          (*
           (* (pow t_m 1.5) (/ (sin k) l))
           (* (* (+ t_2 2.0) (tan k)) (pow t_m 1.5)))
          l))
        (/
         2.0
         (*
          (/ (/ t_m l) (/ (/ l t_m) (* (* t_m (sin k)) (tan k))))
          (+ (+ 1.0 t_2) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.8e-29) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 4.8e+143) {
		tmp = 2.0 / (((pow(t_m, 1.5) * (sin(k) / l)) * (((t_2 + 2.0) * tan(k)) * pow(t_m, 1.5))) / l);
	} else {
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * ((1.0 + t_2) + 1.0));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 6.8d-29) then
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / l)
    else if (t_m <= 4.8d+143) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) * (sin(k) / l)) * (((t_2 + 2.0d0) * tan(k)) * (t_m ** 1.5d0))) / l)
    else
        tmp = 2.0d0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * ((1.0d0 + t_2) + 1.0d0))
    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 t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.8e-29) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / l);
	} else if (t_m <= 4.8e+143) {
		tmp = 2.0 / (((Math.pow(t_m, 1.5) * (Math.sin(k) / l)) * (((t_2 + 2.0) * Math.tan(k)) * Math.pow(t_m, 1.5))) / l);
	} else {
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * Math.sin(k)) * Math.tan(k)))) * ((1.0 + t_2) + 1.0));
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 6.8e-29:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	elif t_m <= 4.8e+143:
		tmp = 2.0 / (((math.pow(t_m, 1.5) * (math.sin(k) / l)) * (((t_2 + 2.0) * math.tan(k)) * math.pow(t_m, 1.5))) / l)
	else:
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * math.sin(k)) * math.tan(k)))) * ((1.0 + t_2) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 6.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 4.8e+143)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 1.5) * Float64(sin(k) / l)) * Float64(Float64(Float64(t_2 + 2.0) * tan(k)) * (t_m ^ 1.5))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) / Float64(Float64(l / t_m) / Float64(Float64(t_m * sin(k)) * tan(k)))) * Float64(Float64(1.0 + t_2) + 1.0)));
	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)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 6.8e-29)
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	elseif (t_m <= 4.8e+143)
		tmp = 2.0 / ((((t_m ^ 1.5) * (sin(k) / l)) * (((t_2 + 2.0) * tan(k)) * (t_m ^ 1.5))) / l);
	else
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * ((1.0 + t_2) + 1.0));
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.8e-29], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+143], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$2 + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.79999999999999945e-29

   1. Initial program 47.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 64.6

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.6%

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

   if 6.79999999999999945e-29 < t < 4.79999999999999959e143

   1. Initial program 71.7%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

   6. Applied rewrites 96.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot {t}^{1.5}\right) \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   8. Applied rewrites 96.6%

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

   if 4.79999999999999959e143 < t

   1. Initial program 61.6%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 92.1

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

   6. Applied rewrites 92.1%

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

Alternative 3: 91.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t\_m}{\ell}}{\frac{\frac{\ell}{t\_m}}{\left(t\_m \cdot \sin k\right) \cdot \tan k}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<= t_m 4.2e-56)
      (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
      (if (<= t_m 3.4e+144)
        (/ 2.0 (* (* (* (/ t_m l) (tan k)) (* (/ (sin k) l) (* t_m t_m))) t_2))
        (/
         2.0
         (* (/ (/ t_m l) (/ (/ l t_m) (* (* t_m (sin k)) (tan k)))) t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if (t_m <= 4.2e-56) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 3.4e+144) {
		tmp = 2.0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * t_2);
	} else {
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * t_2);
	}
	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 = (1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0
    if (t_m <= 4.2d-56) then
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / l)
    else if (t_m <= 3.4d+144) then
        tmp = 2.0d0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * t_2)
    else
        tmp = 2.0d0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * t_2)
    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 = (1.0 + Math.pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if (t_m <= 4.2e-56) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / l);
	} else if (t_m <= 3.4e+144) {
		tmp = 2.0 / ((((t_m / l) * Math.tan(k)) * ((Math.sin(k) / l) * (t_m * t_m))) * t_2);
	} else {
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * Math.sin(k)) * Math.tan(k)))) * t_2);
	}
	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 = (1.0 + math.pow((k / t_m), 2.0)) + 1.0
	tmp = 0
	if t_m <= 4.2e-56:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	elif t_m <= 3.4e+144:
		tmp = 2.0 / ((((t_m / l) * math.tan(k)) * ((math.sin(k) / l) * (t_m * t_m))) * t_2)
	else:
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * math.sin(k)) * math.tan(k)))) * t_2)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (t_m <= 4.2e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 3.4e+144)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * tan(k)) * Float64(Float64(sin(k) / l) * Float64(t_m * t_m))) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) / Float64(Float64(l / t_m) / Float64(Float64(t_m * sin(k)) * tan(k)))) * t_2));
	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 = (1.0 + ((k / t_m) ^ 2.0)) + 1.0;
	tmp = 0.0;
	if (t_m <= 4.2e-56)
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	elseif (t_m <= 3.4e+144)
		tmp = 2.0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * t_2);
	else
		tmp = 2.0 / (((t_m / l) / ((l / t_m) / ((t_m * sin(k)) * tan(k)))) * t_2);
	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[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-56], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+144], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] / N[(N[(l / t$95$m), $MachinePrecision] / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.20000000000000012e-56

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.5%

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

   if 4.20000000000000012e-56 < t < 3.3999999999999999e144

   1. Initial program 69.7%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 93.2

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

   6. Applied rewrites 93.2%

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

   if 3.3999999999999999e144 < t

   1. Initial program 61.6%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 92.1

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

   6. Applied rewrites 92.1%

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

Alternative 4: 90.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell \cdot \frac{\frac{\ell}{t\_m}}{\left(t\_m \cdot \sin k\right) \cdot \tan k}} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
   (*
    t_s
    (if (<= t_m 4.2e-56)
      (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
      (if (<= t_m 3.5e+144)
        (/ 2.0 (* (* (* (/ t_m l) (tan k)) (* (/ (sin k) l) (* t_m t_m))) t_2))
        (/
         2.0
         (* (/ t_m (* l (/ (/ l t_m) (* (* t_m (sin k)) (tan k))))) t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (1.0 + pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if (t_m <= 4.2e-56) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 3.5e+144) {
		tmp = 2.0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * t_2);
	} else {
		tmp = 2.0 / ((t_m / (l * ((l / t_m) / ((t_m * sin(k)) * tan(k))))) * t_2);
	}
	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 = (1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0
    if (t_m <= 4.2d-56) then
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / l)
    else if (t_m <= 3.5d+144) then
        tmp = 2.0d0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * t_2)
    else
        tmp = 2.0d0 / ((t_m / (l * ((l / t_m) / ((t_m * sin(k)) * tan(k))))) * t_2)
    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 = (1.0 + Math.pow((k / t_m), 2.0)) + 1.0;
	double tmp;
	if (t_m <= 4.2e-56) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / l);
	} else if (t_m <= 3.5e+144) {
		tmp = 2.0 / ((((t_m / l) * Math.tan(k)) * ((Math.sin(k) / l) * (t_m * t_m))) * t_2);
	} else {
		tmp = 2.0 / ((t_m / (l * ((l / t_m) / ((t_m * Math.sin(k)) * Math.tan(k))))) * t_2);
	}
	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 = (1.0 + math.pow((k / t_m), 2.0)) + 1.0
	tmp = 0
	if t_m <= 4.2e-56:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	elif t_m <= 3.5e+144:
		tmp = 2.0 / ((((t_m / l) * math.tan(k)) * ((math.sin(k) / l) * (t_m * t_m))) * t_2)
	else:
		tmp = 2.0 / ((t_m / (l * ((l / t_m) / ((t_m * math.sin(k)) * math.tan(k))))) * t_2)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)
	tmp = 0.0
	if (t_m <= 4.2e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 3.5e+144)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * tan(k)) * Float64(Float64(sin(k) / l) * Float64(t_m * t_m))) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / Float64(l * Float64(Float64(l / t_m) / Float64(Float64(t_m * sin(k)) * tan(k))))) * t_2));
	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 = (1.0 + ((k / t_m) ^ 2.0)) + 1.0;
	tmp = 0.0;
	if (t_m <= 4.2e-56)
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	elseif (t_m <= 3.5e+144)
		tmp = 2.0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * t_2);
	else
		tmp = 2.0 / ((t_m / (l * ((l / t_m) / ((t_m * sin(k)) * tan(k))))) * t_2);
	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[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-56], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+144], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / N[(l * N[(N[(l / t$95$m), $MachinePrecision] / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.20000000000000012e-56

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.5%

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

   if 4.20000000000000012e-56 < t < 3.4999999999999998e144

   1. Initial program 69.7%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 93.2

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

   6. Applied rewrites 93.2%

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

   if 3.4999999999999998e144 < t

   1. Initial program 61.6%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 55.3

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

   4. Applied rewrites 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l/ N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 88.6

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

   6. Applied rewrites 88.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t \cdot t\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \frac{\frac{\ell}{t}}{\left(t \cdot \sin k\right) \cdot \tan k}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.45 \cdot 10^{+106}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \tan k\right) \cdot \left(\frac{\sin k}{\ell} \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 4.2e-56)
    (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
    (if (<= t_m 2.45e+106)
      (/
       2.0
       (*
        (* (* (/ t_m l) (tan k)) (* (/ (sin k) l) (* t_m t_m)))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (* (/ l (pow (* k t_m) 2.0)) (/ l 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 (t_m <= 4.2e-56) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 2.45e+106) {
		tmp = 2.0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / 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 (t_m <= 4.2d-56) then
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / l)
    else if (t_m <= 2.45d+106) then
        tmp = 2.0d0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / 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 (t_m <= 4.2e-56) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / l);
	} else if (t_m <= 2.45e+106) {
		tmp = 2.0 / ((((t_m / l) * Math.tan(k)) * ((Math.sin(k) / l) * (t_m * t_m))) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / 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 t_m <= 4.2e-56:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	elif t_m <= 2.45e+106:
		tmp = 2.0 / ((((t_m / l) * math.tan(k)) * ((math.sin(k) / l) * (t_m * t_m))) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / 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 (t_m <= 4.2e-56)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 2.45e+106)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * tan(k)) * Float64(Float64(sin(k) / l) * Float64(t_m * t_m))) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / 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 (t_m <= 4.2e-56)
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	elseif (t_m <= 2.45e+106)
		tmp = 2.0 / ((((t_m / l) * tan(k)) * ((sin(k) / l) * (t_m * t_m))) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / 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[t$95$m, 4.2e-56], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.45e+106], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.20000000000000012e-56

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 64.8

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

   5. Applied rewrites 64.8%

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.5%

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

   if 4.20000000000000012e-56 < t < 2.44999999999999999e106

   1. Initial program 73.2%

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

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.3

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

   6. Applied rewrites 94.3%

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

   if 2.44999999999999999e106 < t

   1. Initial program 61.3%

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

   3. Taylor expanded in k around 0

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

      1. 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 50.8

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

   5. Applied rewrites 50.8%

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

   7. Applied rewrites 50.8%

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

   9. Applied rewrites 59.7%

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

   11. Applied rewrites 84.7%

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

Alternative 6: 86.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.3 \cdot 10^{+104}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{\sin k}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 6.8e-29)
    (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
    (if (<= t_m 1.3e+104)
      (/
       2.0
       (*
        (* (* t_m (* (/ (* t_m t_m) l) (/ (sin k) l))) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (* (/ l (pow (* k t_m) 2.0)) (/ l 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 (t_m <= 6.8e-29) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 1.3e+104) {
		tmp = 2.0 / (((t_m * (((t_m * t_m) / l) * (sin(k) / l))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / 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 (t_m <= 6.8d-29) then
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / l)
    else if (t_m <= 1.3d+104) then
        tmp = 2.0d0 / (((t_m * (((t_m * t_m) / l) * (sin(k) / l))) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / 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 (t_m <= 6.8e-29) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / l);
	} else if (t_m <= 1.3e+104) {
		tmp = 2.0 / (((t_m * (((t_m * t_m) / l) * (Math.sin(k) / l))) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / 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 t_m <= 6.8e-29:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	elif t_m <= 1.3e+104:
		tmp = 2.0 / (((t_m * (((t_m * t_m) / l) * (math.sin(k) / l))) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / 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 (t_m <= 6.8e-29)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 1.3e+104)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(Float64(t_m * t_m) / l) * Float64(sin(k) / l))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / 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 (t_m <= 6.8e-29)
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	elseif (t_m <= 1.3e+104)
		tmp = 2.0 / (((t_m * (((t_m * t_m) / l) * (sin(k) / l))) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / 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[t$95$m, 6.8e-29], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.3e+104], N[(2.0 / N[(N[(N[(t$95$m * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.79999999999999945e-29

   1. Initial program 47.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 64.6

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.6%

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

   if 6.79999999999999945e-29 < t < 1.3e104

   1. Initial program 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-pow.f64 N/A

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

      7. cube-mult N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 93.1

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

   4. Applied rewrites 93.1%

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

   if 1.3e104 < t

   1. Initial program 61.3%

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

   3. Taylor expanded in k around 0

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

      1. 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 50.8

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

   5. Applied rewrites 50.8%

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

   7. Applied rewrites 50.8%

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

   9. Applied rewrites 59.7%

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

   11. Applied rewrites 84.7%

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

Alternative 7: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(\sin k \cdot k\right)}^{2}}{\ell} \cdot t\_m}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \sin k\right) \cdot {\ell}^{-2}\right) \cdot k\right) \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.1e-41)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (if (<= k 6.2e+151)
      (/ 2.0 (/ (* (/ (pow (* (sin k) k) 2.0) l) t_m) (* (cos k) l)))
      (/ 2.0 (* (* (* (* (tan k) (sin k)) (pow l -2.0)) k) (* 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) {
	double tmp;
	if (k <= 1.1e-41) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 6.2e+151) {
		tmp = 2.0 / (((pow((sin(k) * k), 2.0) / l) * t_m) / (cos(k) * l));
	} else {
		tmp = 2.0 / ((((tan(k) * sin(k)) * pow(l, -2.0)) * k) * (k * 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 <= 1.1d-41) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else if (k <= 6.2d+151) then
        tmp = 2.0d0 / (((((sin(k) * k) ** 2.0d0) / l) * t_m) / (cos(k) * l))
    else
        tmp = 2.0d0 / ((((tan(k) * sin(k)) * (l ** (-2.0d0))) * k) * (k * 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 <= 1.1e-41) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 6.2e+151) {
		tmp = 2.0 / (((Math.pow((Math.sin(k) * k), 2.0) / l) * t_m) / (Math.cos(k) * l));
	} else {
		tmp = 2.0 / ((((Math.tan(k) * Math.sin(k)) * Math.pow(l, -2.0)) * k) * (k * 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 <= 1.1e-41:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	elif k <= 6.2e+151:
		tmp = 2.0 / (((math.pow((math.sin(k) * k), 2.0) / l) * t_m) / (math.cos(k) * l))
	else:
		tmp = 2.0 / ((((math.tan(k) * math.sin(k)) * math.pow(l, -2.0)) * k) * (k * 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 <= 1.1e-41)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	elseif (k <= 6.2e+151)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k) * k) ^ 2.0) / l) * t_m) / Float64(cos(k) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) * sin(k)) * (l ^ -2.0)) * k) * Float64(k * 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 <= 1.1e-41)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	elseif (k <= 6.2e+151)
		tmp = 2.0 / (((((sin(k) * k) ^ 2.0) / l) * t_m) / (cos(k) * l));
	else
		tmp = 2.0 / ((((tan(k) * sin(k)) * (l ^ -2.0)) * k) * (k * 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, 1.1e-41], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.2e+151], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.1e-41

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.8%

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

   if 1.1e-41 < k < 6.2000000000000004e151

   1. Initial program 53.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 88.5%

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

   if 6.2000000000000004e151 < k

   1. Initial program 33.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 79.5%

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

Alternative 8: 75.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-39)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-39) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / 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 (k <= 1.7d-39) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / 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 (k <= 1.7e-39) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / 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 k <= 1.7e-39:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	else:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.7e-39)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / 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 (k <= 1.7e-39)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	else
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / 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[k, 1.7e-39], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e-39

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.8%

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

   if 1.7e-39 < k

   1. Initial program 44.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.6

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.6%

       \[\leadsto \frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot t\_m\right) \cdot \left(\left(\tan k \cdot \sin k\right) \cdot {\ell}^{-2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-39)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (/ 2.0 (* k (* (* k t_m) (* (* (tan k) (sin k)) (pow 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 <= 1.7e-39) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (k * ((k * t_m) * ((tan(k) * sin(k)) * pow(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 <= 1.7d-39) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else
        tmp = 2.0d0 / (k * ((k * t_m) * ((tan(k) * sin(k)) * (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 <= 1.7e-39) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (k * ((k * t_m) * ((Math.tan(k) * Math.sin(k)) * Math.pow(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 <= 1.7e-39:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	else:
		tmp = 2.0 / (k * ((k * t_m) * ((math.tan(k) * math.sin(k)) * math.pow(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 <= 1.7e-39)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(k * t_m) * Float64(Float64(tan(k) * sin(k)) * (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 <= 1.7e-39)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	else
		tmp = 2.0 / (k * ((k * t_m) * ((tan(k) * sin(k)) * (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, 1.7e-39], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(k * t$95$m), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e-39

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.8%

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

   if 1.7e-39 < k

   1. Initial program 44.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.6

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 76.3%

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

Alternative 10: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.45 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot k\right) \cdot k\right) \cdot \frac{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t\_m \leq 1.85 \cdot 10^{+83}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t\_m \cdot t\_m}}\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 1.45e-78)
    (/
     2.0
     (*
      (* (* t_m k) k)
      (/ (- (/ 0.5 l) (/ (* 0.5 (cos (+ k k))) l)) (* (cos k) l))))
    (if (<= t_m 1.85e+83)
      (/
       2.0
       (*
        (* (/ t_m l) (/ (* (tan k) (sin k)) (/ l (* t_m t_m))))
        (fma (/ k t_m) (/ k t_m) 2.0)))
      (* (/ l (pow (* k t_m) 2.0)) (/ l 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 (t_m <= 1.45e-78) {
		tmp = 2.0 / (((t_m * k) * k) * (((0.5 / l) - ((0.5 * cos((k + k))) / l)) / (cos(k) * l)));
	} else if (t_m <= 1.85e+83) {
		tmp = 2.0 / (((t_m / l) * ((tan(k) * sin(k)) / (l / (t_m * t_m)))) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / 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 (t_m <= 1.45e-78)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * k) * k) * Float64(Float64(Float64(0.5 / l) - Float64(Float64(0.5 * cos(Float64(k + k))) / l)) / Float64(cos(k) * l))));
	elseif (t_m <= 1.85e+83)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(tan(k) * sin(k)) / Float64(l / Float64(t_m * t_m)))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.45e-78], N[(2.0 / N[(N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(0.5 / l), $MachinePrecision] - N[(N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.85e+83], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.45e-78

   1. Initial program 46.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 53.8%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}{\color{blue}{\cos k} \cdot \ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.4%

      \[\leadsto \frac{2}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\color{blue}{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}}{\cos k \cdot \ell}} \]

   if 1.45e-78 < t < 1.8500000000000001e83

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 79.8

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

   4. Applied rewrites 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 79.8

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

   6. Applied rewrites 79.8%

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

   if 1.8500000000000001e83 < t

   1. Initial program 60.8%

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

   3. Taylor expanded in k around 0

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

      1. 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

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 50.9%

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

   9. Applied rewrites 59.3%

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

   11. Applied rewrites 83.0%

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

Alternative 11: 73.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot k\right) \cdot k\right) \cdot \frac{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e-8)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (/
     2.0
     (*
      (* (* t_m k) k)
      (/ (- (/ 0.5 l) (/ (* 0.5 (cos (+ k k))) l)) (* (cos k) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-8) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((t_m * k) * k) * (((0.5 / l) - ((0.5 * cos((k + k))) / l)) / (cos(k) * 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 (k <= 1.35d-8) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else
        tmp = 2.0d0 / (((t_m * k) * k) * (((0.5d0 / l) - ((0.5d0 * cos((k + k))) / l)) / (cos(k) * 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 (k <= 1.35e-8) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((t_m * k) * k) * (((0.5 / l) - ((0.5 * Math.cos((k + k))) / l)) / (Math.cos(k) * 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 k <= 1.35e-8:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	else:
		tmp = 2.0 / (((t_m * k) * k) * (((0.5 / l) - ((0.5 * math.cos((k + k))) / l)) / (math.cos(k) * l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e-8)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * k) * k) * Float64(Float64(Float64(0.5 / l) - Float64(Float64(0.5 * cos(Float64(k + k))) / l)) / Float64(cos(k) * 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 (k <= 1.35e-8)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	else
		tmp = 2.0 / (((t_m * k) * k) * (((0.5 / l) - ((0.5 * cos((k + k))) / l)) / (cos(k) * 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[k, 1.35e-8], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(0.5 / l), $MachinePrecision] - N[(N[(0.5 * N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000001e-8

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.6%

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

   if 1.35000000000000001e-8 < k

   1. Initial program 44.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}{\color{blue}{\cos k} \cdot \ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

      \[\leadsto \frac{2}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\color{blue}{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}}{\cos k \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{\frac{\mathsf{fma}\left(-0.5, \cos \left(2 \cdot k\right), 0.5\right)}{\ell}}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e-8)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (/
     2.0
     (*
      (* (* k k) t_m)
      (/ (/ (fma -0.5 (cos (* 2.0 k)) 0.5) l) (* (cos k) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-8) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((k * k) * t_m) * ((fma(-0.5, cos((2.0 * k)), 0.5) / l) / (cos(k) * l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e-8)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(fma(-0.5, cos(Float64(2.0 * k)), 0.5) / l) / Float64(cos(k) * l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.35e-8], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(-0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000001e-8

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.6%

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

   if 1.35000000000000001e-8 < k

   1. Initial program 44.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}{\color{blue}{\cos k} \cdot \ell}} \]

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 68.2%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{\mathsf{fma}\left(-0.5, \cos \left(2 \cdot k\right), 0.5\right)}{\ell}}{\color{blue}{\cos k} \cdot \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 72.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{\mathsf{fma}\left(-0.5, \cos \left(2 \cdot k\right), 0.5\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e-8)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (/
     2.0
     (*
      (* (* k k) t_m)
      (/ (fma -0.5 (cos (* 2.0 k)) 0.5) (* (* (cos k) l) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-8) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((k * k) * t_m) * (fma(-0.5, cos((2.0 * k)), 0.5) / ((cos(k) * l) * l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e-8)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(fma(-0.5, cos(Float64(2.0 * k)), 0.5) / Float64(Float64(cos(k) * l) * l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.35e-8], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(-0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000001e-8

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.6%

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

   if 1.35000000000000001e-8 < k

   1. Initial program 44.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{0.5}{\ell} - \frac{0.5 \cdot \cos \left(k + k\right)}{\ell}}{\color{blue}{\cos k} \cdot \ell}} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 68.1%

       \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.5, \cos \left(2 \cdot k\right), 0.5\right)}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.7e-39)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
    (/ 2.0 (* (* (* k k) t_m) (/ (/ (* k k) l) (* (cos k) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.7e-39) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * 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 (k <= 1.7d-39) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else
        tmp = 2.0d0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * 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 (k <= 1.7e-39) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else {
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (Math.cos(k) * 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 k <= 1.7e-39:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	else:
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (math.cos(k) * l)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.7e-39)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(Float64(k * k) / l) / Float64(cos(k) * 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 (k <= 1.7e-39)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	else
		tmp = 2.0 / (((k * k) * t_m) * (((k * k) / l) / (cos(k) * 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[k, 1.7e-39], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.7e-39

   1. Initial program 53.7%

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

   3. Taylor expanded in k around 0

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

      1. 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 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 57.0%

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

   11. Applied rewrites 70.8%

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

   if 1.7e-39 < k

   1. Initial program 44.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 63.0%

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

Alternative 15: 74.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\ell}{{t\_m}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 5.4e-110)
    (/ 2.0 (* (/ (pow k 4.0) l) (/ t_m l)))
    (if (<= t_m 4.9e+32)
      (* (/ l (pow t_m 3.0)) (/ (/ l k) k))
      (* (/ l (pow (* k t_m) 2.0)) (/ l 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 (t_m <= 5.4e-110) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t_m / l));
	} else if (t_m <= 4.9e+32) {
		tmp = (l / pow(t_m, 3.0)) * ((l / k) / k);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / 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 (t_m <= 5.4d-110) then
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t_m / l))
    else if (t_m <= 4.9d+32) then
        tmp = (l / (t_m ** 3.0d0)) * ((l / k) / k)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / 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 (t_m <= 5.4e-110) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t_m / l));
	} else if (t_m <= 4.9e+32) {
		tmp = (l / Math.pow(t_m, 3.0)) * ((l / k) / k);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / 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 t_m <= 5.4e-110:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t_m / l))
	elif t_m <= 4.9e+32:
		tmp = (l / math.pow(t_m, 3.0)) * ((l / k) / k)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / 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 (t_m <= 5.4e-110)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t_m / l)));
	elseif (t_m <= 4.9e+32)
		tmp = Float64(Float64(l / (t_m ^ 3.0)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / 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 (t_m <= 5.4e-110)
		tmp = 2.0 / (((k ^ 4.0) / l) * (t_m / l));
	elseif (t_m <= 4.9e+32)
		tmp = (l / (t_m ^ 3.0)) * ((l / k) / k);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / 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[t$95$m, 5.4e-110], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.9e+32], N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999996e-110

   1. Initial program 45.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 5.3999999999999996e-110 < t < 4.9000000000000001e32

   1. Initial program 73.9%

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

   3. Taylor expanded in k around 0

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

      1. 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 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 72.5%

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

   if 4.9000000000000001e32 < 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. Add Preprocessing

   3. Taylor expanded in k around 0

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

      1. 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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 52.0%

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

   9. Applied rewrites 58.4%

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

   11. Applied rewrites 78.8%

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

Alternative 16: 72.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}{\ell}, 0.08611111111111111, t\_2 \cdot 0.16666666666666666\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;t\_m \leq 4.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\ell}{{t\_m}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (* t_m t_m) l)))
   (*
    t_s
    (if (<= t_m 5.4e-110)
      (/
       2.0
       (*
        (*
         (/ t_m l)
         (*
          (fma
           (*
            (fma
             (/ (* (* (* k k) t_m) t_m) l)
             0.08611111111111111
             (* t_2 0.16666666666666666))
            k)
           k
           t_2)
          (* k k)))
        2.0))
      (if (<= t_m 4.9e+32)
        (* (/ l (pow t_m 3.0)) (/ (/ l k) k))
        (* (/ l (pow (* k t_m) 2.0)) (/ l 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 5.4e-110) {
		tmp = 2.0 / (((t_m / l) * (fma((fma(((((k * k) * t_m) * t_m) / l), 0.08611111111111111, (t_2 * 0.16666666666666666)) * k), k, t_2) * (k * k))) * 2.0);
	} else if (t_m <= 4.9e+32) {
		tmp = (l / pow(t_m, 3.0)) * ((l / k) / k);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / 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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 5.4e-110)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(fma(Float64(fma(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) / l), 0.08611111111111111, Float64(t_2 * 0.16666666666666666)) * k), k, t_2) * Float64(k * k))) * 2.0));
	elseif (t_m <= 4.9e+32)
		tmp = Float64(Float64(l / (t_m ^ 3.0)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.4e-110], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 0.08611111111111111 + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.9e+32], N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999996e-110

   1. Initial program 45.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. cube-mult N/A

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

      10. times-frac N/A

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

      11. times-frac N/A

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

      12. clear-num N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 51.2

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right)\right)}\right) \cdot 2} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

   10. Applied rewrites 58.7%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell}, 0.08611111111111111, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666\right) \cdot k, k, \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}\right) \cdot 2} \]

   if 5.3999999999999996e-110 < t < 4.9000000000000001e32

   1. Initial program 73.9%

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

   3. Taylor expanded in k around 0

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

      1. 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 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 72.5%

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

   if 4.9000000000000001e32 < 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. Add Preprocessing

   3. Taylor expanded in k around 0

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

      1. 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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 52.0%

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

   9. Applied rewrites 58.4%

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

   11. Applied rewrites 78.8%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 72.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}{\ell}, 0.08611111111111111, t\_2 \cdot 0.16666666666666666\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;t\_m \leq 4.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (/ (* t_m t_m) l)))
   (*
    t_s
    (if (<= t_m 5.4e-110)
      (/
       2.0
       (*
        (*
         (/ t_m l)
         (*
          (fma
           (*
            (fma
             (/ (* (* (* k k) t_m) t_m) l)
             0.08611111111111111
             (* t_2 0.16666666666666666))
            k)
           k
           t_2)
          (* k k)))
        2.0))
      (if (<= t_m 4.9e+32)
        (* (/ l (* (* t_m t_m) t_m)) (/ (/ l k) k))
        (* (/ l (pow (* k t_m) 2.0)) (/ l 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 t_2 = (t_m * t_m) / l;
	double tmp;
	if (t_m <= 5.4e-110) {
		tmp = 2.0 / (((t_m / l) * (fma((fma(((((k * k) * t_m) * t_m) / l), 0.08611111111111111, (t_2 * 0.16666666666666666)) * k), k, t_2) * (k * k))) * 2.0);
	} else if (t_m <= 4.9e+32) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / 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)
	t_2 = Float64(Float64(t_m * t_m) / l)
	tmp = 0.0
	if (t_m <= 5.4e-110)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(fma(Float64(fma(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) / l), 0.08611111111111111, Float64(t_2 * 0.16666666666666666)) * k), k, t_2) * Float64(k * k))) * 2.0));
	elseif (t_m <= 4.9e+32)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.4e-110], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 0.08611111111111111 + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.9e+32], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 5.3999999999999996e-110

   1. Initial program 45.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. clear-num N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{{t}^{3}}}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{{t}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{\color{blue}{{t}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. cube-mult N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot t\right)}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{\frac{\ell}{t}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. clear-num N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\color{blue}{\frac{\ell}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. lower-*.f64 51.2

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{\color{blue}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 51.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.9%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right) \cdot \color{blue}{2}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right)\right)}\right) \cdot 2} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

   10. Applied rewrites 58.7%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell}, 0.08611111111111111, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666\right) \cdot k, k, \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}\right) \cdot 2} \]

   if 5.3999999999999996e-110 < t < 4.9000000000000001e32

   1. Initial program 73.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 66.6

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

   if 4.9000000000000001e32 < 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. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 52.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.4%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 78.8%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot t\_m}{\ell}\\ t_3 := \left(k \cdot k\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t\_3 \cdot t\_m}{\ell}, 0.08611111111111111, t\_2 \cdot 0.16666666666666666\right) \cdot k, k, t\_2\right) \cdot \left(k \cdot k\right)\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

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 (/ (* t_m t_m) l)) (t_3 (* (* k k) t_m)))
   (*
    t_s
    (if (<= k 7.5e-142)
      (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
      (if (<= k 4.5e-78)
        (/ (/ (* (/ l t_m) l) t_m) t_3)
        (/
         2.0
         (*
          (*
           (/ t_m l)
           (*
            (fma
             (*
              (fma
               (/ (* t_3 t_m) l)
               0.08611111111111111
               (* t_2 0.16666666666666666))
              k)
             k
             t_2)
            (* k k)))
          2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (t_m * t_m) / l;
	double t_3 = (k * k) * t_m;
	double tmp;
	if (k <= 7.5e-142) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else if (k <= 4.5e-78) {
		tmp = (((l / t_m) * l) / t_m) / t_3;
	} else {
		tmp = 2.0 / (((t_m / l) * (fma((fma(((t_3 * t_m) / l), 0.08611111111111111, (t_2 * 0.16666666666666666)) * k), k, t_2) * (k * k))) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(t_m * t_m) / l)
	t_3 = Float64(Float64(k * k) * t_m)
	tmp = 0.0
	if (k <= 7.5e-142)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	elseif (k <= 4.5e-78)
		tmp = Float64(Float64(Float64(Float64(l / t_m) * l) / t_m) / t_3);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(fma(Float64(fma(Float64(Float64(t_3 * t_m) / l), 0.08611111111111111, Float64(t_2 * 0.16666666666666666)) * k), k, t_2) * Float64(k * k))) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 7.5e-142], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.5e-78], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$3), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(t$95$3 * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 0.08611111111111111 + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k + t$95$2), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.49999999999999958e-142

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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 51.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.8%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 7.49999999999999958e-142 < k < 4.5e-78

   1. Initial program 63.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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 70.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.8%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 84.1%

       \[\leadsto \frac{\frac{\frac{\ell}{t} \cdot \ell}{t}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   if 4.5e-78 < k

   1. Initial program 46.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. clear-num N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{{t}^{3}}}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{{t}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{\color{blue}{{t}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. cube-mult N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot t\right)}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{\frac{\ell}{t}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. clear-num N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\color{blue}{\frac{\ell}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. lower-*.f64 54.1

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{\color{blue}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 54.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.0%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right) \cdot \color{blue}{2}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right)\right)}\right) \cdot 2} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left({k}^{2} \cdot \left(\frac{31}{360} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{1}{6} \cdot \frac{{t}^{2}}{\ell}\right) + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

   10. Applied rewrites 56.5%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell}, 0.08611111111111111, \frac{t \cdot t}{\ell} \cdot 0.16666666666666666\right) \cdot k, k, \frac{t \cdot t}{\ell}\right) \cdot \left(k \cdot k\right)\right)}\right) \cdot 2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 65.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(\left(\mathsf{fma}\left(\frac{t\_2 \cdot t\_m}{\ell}, 0.16666666666666666, \frac{t\_m \cdot t\_m}{\ell}\right) \cdot k\right) \cdot k\right)\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

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 (* (* k k) t_m)))
   (*
    t_s
    (if (<= k 7.5e-142)
      (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
      (if (<= k 2.4e-87)
        (/ (/ (* (/ l t_m) l) t_m) t_2)
        (/
         2.0
         (*
          (*
           (/ t_m l)
           (*
            (* (fma (/ (* t_2 t_m) l) 0.16666666666666666 (/ (* t_m t_m) l)) k)
            k))
          2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * t_m;
	double tmp;
	if (k <= 7.5e-142) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else if (k <= 2.4e-87) {
		tmp = (((l / t_m) * l) / t_m) / t_2;
	} else {
		tmp = 2.0 / (((t_m / l) * ((fma(((t_2 * t_m) / l), 0.16666666666666666, ((t_m * t_m) / l)) * k) * k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * k) * t_m)
	tmp = 0.0
	if (k <= 7.5e-142)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	elseif (k <= 2.4e-87)
		tmp = Float64(Float64(Float64(Float64(l / t_m) * l) / t_m) / t_2);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(Float64(t_2 * t_m) / l), 0.16666666666666666, Float64(Float64(t_m * t_m) / l)) * k) * k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 7.5e-142], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.4e-87], N[(N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 0.16666666666666666 + N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 7.49999999999999958e-142

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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 51.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.8%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 7.49999999999999958e-142 < k < 2.4e-87

   1. Initial program 62.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 63.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.5%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.2%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.1%

       \[\leadsto \frac{\frac{\frac{\ell}{t} \cdot \ell}{t}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   if 2.4e-87 < k

   1. Initial program 47.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. clear-num N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{{t}^{3}}}} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{{t}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{\color{blue}{{t}^{3}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. cube-mult N/A

         \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(t \cdot t\right)}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\frac{1 \cdot \left(\sin k \cdot \tan k\right)}{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{\frac{\ell}{t}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. clear-num N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{\sin k \cdot \tan k}{\frac{\ell}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{\tan k \cdot \sin k}}{\frac{\ell}{t \cdot t}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      18. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\color{blue}{\frac{\ell}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      19. lower-*.f64 54.6

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{\color{blue}{t \cdot t}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 54.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right) \cdot \color{blue}{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.6%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\tan k \cdot \sin k}{\frac{\ell}{t \cdot t}}\right) \cdot \color{blue}{2}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left({k}^{2} \cdot \left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{t}^{2}}{\ell}\right)\right)}\right) \cdot 2} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{t}^{2}}{\ell}\right) \cdot {k}^{2}\right)}\right) \cdot 2} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{t}^{2}}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{t}^{2}}{\ell}\right) \cdot k\right) \cdot k\right)}\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\left(\frac{1}{6} \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell} + \frac{{t}^{2}}{\ell}\right) \cdot k\right) \cdot k\right)}\right) \cdot 2} \]

   10. Applied rewrites 55.6%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell}, 0.16666666666666666, \frac{t \cdot t}{\ell}\right) \cdot k\right) \cdot k\right)}\right) \cdot 2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 64.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 5.3e-110)
    (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m)))
    (if (<= t_m 2.5e+106)
      (* (/ l (* (* t_m t_m) t_m)) (/ (/ l k) k))
      (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.3e-110) {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	} else if (t_m <= 2.5e+106) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.3d-110) then
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
    else if (t_m <= 2.5d+106) then
        tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
    else
        tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.3e-110) {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	} else if (t_m <= 2.5e+106) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.3e-110:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
	elif t_m <= 2.5e+106:
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
	else:
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.3e-110)
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m)));
	elseif (t_m <= 2.5e+106)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.3e-110)
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	elseif (t_m <= 2.5e+106)
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	else
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.3e-110], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+106], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.30000000000000001e-110

   1. Initial program 45.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 45.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.7%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.9%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.7%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

   if 5.30000000000000001e-110 < t < 2.4999999999999999e106

   1. Initial program 71.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 61.3

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.3%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.6%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\frac{\ell}{k}}{\color{blue}{k}} \]

   if 2.4999999999999999e106 < t

   1. Initial program 60.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 52.2

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.2%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.4%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.7%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 64.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{t\_m \cdot t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+84}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \end{array} \]

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 (/ l (* t_m t_m))))
   (*
    t_s
    (if (<= t_m 2.3e-169)
      (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m)))
      (if (<= t_m 1.5e+84)
        (* t_2 (/ (/ l (* k k)) t_m))
        (/ (* t_2 l) (* (* k t_m) k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (t_m * t_m);
	double tmp;
	if (t_m <= 2.3e-169) {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	} else if (t_m <= 1.5e+84) {
		tmp = t_2 * ((l / (k * k)) / t_m);
	} else {
		tmp = (t_2 * l) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (t_m * t_m)
    if (t_m <= 2.3d-169) then
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
    else if (t_m <= 1.5d+84) then
        tmp = t_2 * ((l / (k * k)) / t_m)
    else
        tmp = (t_2 * l) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / (t_m * t_m);
	double tmp;
	if (t_m <= 2.3e-169) {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	} else if (t_m <= 1.5e+84) {
		tmp = t_2 * ((l / (k * k)) / t_m);
	} else {
		tmp = (t_2 * l) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = l / (t_m * t_m)
	tmp = 0
	if t_m <= 2.3e-169:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
	elif t_m <= 1.5e+84:
		tmp = t_2 * ((l / (k * k)) / t_m)
	else:
		tmp = (t_2 * l) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(l / Float64(t_m * t_m))
	tmp = 0.0
	if (t_m <= 2.3e-169)
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m)));
	elseif (t_m <= 1.5e+84)
		tmp = Float64(t_2 * Float64(Float64(l / Float64(k * k)) / t_m));
	else
		tmp = Float64(Float64(t_2 * l) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = l / (t_m * t_m);
	tmp = 0.0;
	if (t_m <= 2.3e-169)
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	elseif (t_m <= 1.5e+84)
		tmp = t_2 * ((l / (k * k)) / t_m);
	else
		tmp = (t_2 * l) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.3e-169], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+84], N[(t$95$2 * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.3000000000000001e-169

   1. Initial program 47.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 48.3

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.3%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.6%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.4%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]

   if 2.3000000000000001e-169 < t < 1.49999999999999998e84

   1. Initial program 57.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.1%

      \[\leadsto \frac{\ell}{t \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{t}} \]

   if 1.49999999999999998e84 < t

   1. Initial program 60.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 72.5%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 66.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 2.35e-142)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* 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 <= 2.35e-142) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 2.35d-142) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 2.35e-142) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 2.35e-142:
		tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 2.35e-142)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / 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 <= 2.35e-142)
		tmp = ((l / (t_m * t_m)) / k) * (l / (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, 2.35e-142], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.34999999999999995e-142

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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 51.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.8%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 2.34999999999999995e-142 < k

   1. Initial program 49.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 44.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.6%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.6%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 65.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.8e-142)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (/ (* (/ l t_m) l) (* t_m (* (* k 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) {
	double tmp;
	if (k <= 1.8e-142) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 1.8d-142) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    else
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 1.8e-142) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 1.8e-142:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	else:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 1.8e-142)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * 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 <= 1.8e-142)
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	else
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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, 1.8e-142], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-142

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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 51.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.8%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]

   if 1.8e-142 < k

   1. Initial program 49.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 44.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.6%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.2%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 62.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 8.6e-157)
    (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))
    (/ (* (/ l t_m) l) (* t_m (* (* k 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) {
	double tmp;
	if (k <= 8.6e-157) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 8.6d-157) then
        tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
    else
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 8.6e-157) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 8.6e-157:
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
	else:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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 <= 8.6e-157)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * 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 <= 8.6e-157)
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	else
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * 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, 8.6e-157], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.5999999999999995e-157

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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 50.6

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.5%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.4%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]

   if 8.5999999999999995e-157 < k

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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 45.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.7%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.7%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 61.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

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 t_m) l) (* t_m (* (* k 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 / t_m) * l) / (t_m * ((k * 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 / t_m) * l) / (t_m * ((k * 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 / t_m) * l) / (t_m * ((k * 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 / t_m) * l) / (t_m * ((k * 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 / t_m) * l) / Float64(t_m * Float64(Float64(k * 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 / t_m) * l) / (t_m * ((k * 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 / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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 49.0

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 49.0%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 52.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 56.6%

    \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Alternative 26: 58.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m \cdot t\_m} \end{array} \]

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 (/ 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 * Float64(l / 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 * N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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 49.0

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 49.0%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 52.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 52.9%

    \[\leadsto \frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{\color{blue}{t \cdot t}} \]
12. Add Preprocessing

Alternative 27: 58.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{\left(k \cdot k\right) \cdot t\_m}\right) \end{array} \]

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 (* t_m t_m)) (* (* k 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 * ((l / (t_m * t_m)) / ((k * 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 * ((l / (t_m * t_m)) / ((k * 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 * ((l / (t_m * t_m)) / ((k * 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 * ((l / (t_m * t_m)) / ((k * 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(l * Float64(Float64(l / Float64(t_m * t_m)) / Float64(Float64(k * 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 * ((l / (t_m * t_m)) / ((k * 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[(l * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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 49.0

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 49.0%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 52.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 52.5%

    \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{t \cdot t}}{\left(k \cdot k\right) \cdot t}} \]
12. Add Preprocessing

Alternative 28: 53.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)} \end{array} \]

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 51.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. 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 49.0

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 49.0%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 49.0%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 52.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 48.6%

    \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(t \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))))