Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 86.4%

Time: 16.0s

Alternatives: 23

Speedup: 8.7×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(2, \frac{\frac{{t\_m}^{3}}{k}}{k}, t\_m\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell}\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-34)
    (/
     2.0
     (*
      (*
       (*
        (fma 2.0 (/ (/ (pow t_m 3.0) k) k) t_m)
        (/ (/ (pow (sin k) 2.0) l) (* (cos k) l)))
       k)
      k))
    (/
     2.0
     (*
      (* (* (* (/ t_m l) t_m) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k))) (tan k))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.8e-34) {
		tmp = 2.0 / (((fma(2.0, ((pow(t_m, 3.0) / k) / k), t_m) * ((pow(sin(k), 2.0) / l) / (cos(k) * l))) * k) * k);
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * ((pow((k / t_m), 2.0) + 2.0) * sin(k))) * tan(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.8e-34)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(2.0, Float64(Float64((t_m ^ 3.0) / k) / k), t_m) * Float64(Float64((sin(k) ^ 2.0) / l) / Float64(cos(k) * l))) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) * tan(k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.8e-34], N[(2.0 / N[(N[(N[(N[(2.0 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] + t$95$m), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999982e-34

   1. Initial program 45.9%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 70.8%

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

   if 7.79999999999999982e-34 < t

   1. Initial program 59.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 79.8

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

   4. Applied rewrites 79.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 82.9

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 82.9%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 86.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\left(\left(\mathsf{fma}\left(2, \frac{\frac{{t}^{3}}{k}}{k}, t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k \cdot \ell}\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot 2\right) \cdot \frac{\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}{\ell} \cdot 2\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
       5e+202)
    (/ 2.0 (* (* (* k 2.0) (/ (* (/ (* t_m t_m) l) t_m) l)) k))
    (/ 2.0 (* (* (/ (* (* (* k k) t_m) t_m) l) 2.0) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))))) <= 5e+202) {
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * 2.0) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))) <= 5d+202) then
        tmp = 2.0d0 / (((k * 2.0d0) * ((((t_m * t_m) / l) * t_m) / l)) * k)
    else
        tmp = 2.0d0 / ((((((k * k) * t_m) * t_m) / l) * 2.0d0) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))))) <= 5e+202) {
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	} else {
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * 2.0) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))) <= 5e+202:
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k)
	else:
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * 2.0) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))))) <= 5e+202)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * 2.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_m) / l)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) / l) * 2.0) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))))) <= 5e+202)
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	else
		tmp = 2.0 / ((((((k * k) * t_m) * t_m) / l) * 2.0) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+202], N[(2.0 / N[(N[(N[(k * 2.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.9999999999999999e202

   1. Initial program 77.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 75.5%

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

   9. Applied rewrites 75.5%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 19.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 32.1

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

   4. Applied rewrites 32.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 35.9

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 35.9%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 58.7%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 46.1

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

   11. Applied rewrites 46.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot 2\right) \cdot \frac{\frac{t \cdot t}{\ell} \cdot t}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t\_m}{\ell} \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
       5e+202)
    (/ 2.0 (* (* (* (/ t_m (* l l)) (* t_m t_m)) (* k 2.0)) k))
    (/ 2.0 (* (* (/ (* (/ t_m l) t_m) l) t_m) (* (* 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 tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))))) <= 5e+202) {
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) / l) * t_m) * ((k * k) * 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 ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))) <= 5d+202) then
        tmp = 2.0d0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0d0)) * k)
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) / l) * t_m) * ((k * k) * 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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))))) <= 5e+202) {
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) / l) * t_m) * ((k * k) * 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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))) <= 5e+202:
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k)
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) / l) * t_m) * ((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)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))))) <= 5e+202)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * Float64(t_m * t_m)) * Float64(k * 2.0)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) / l) * t_m) * Float64(Float64(k * k) * 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))))) <= 5e+202)
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k);
	else
		tmp = 2.0 / (((((t_m / l) * t_m) / l) * t_m) * ((k * k) * 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[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+202], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.9999999999999999e202

   1. Initial program 77.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 75.5%

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

   9. Applied rewrites 74.5%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 19.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 32.0

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

   5. Applied rewrites 32.0%

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

   7. Applied rewrites 35.3%

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

   9. Applied rewrites 43.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{t}{\ell} \cdot t}{\ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\frac{\ell}{t\_m} \cdot \ell} \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
       5e+202)
    (/ 2.0 (* (* (* (/ t_m (* l l)) (* t_m t_m)) (* k 2.0)) k))
    (/ 2.0 (* (* (/ t_m (* (/ l t_m) l)) t_m) (* (* 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 tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + pow((k / t_m), 2.0))))) <= 5e+202) {
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * k) * 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 ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))) <= 5d+202) then
        tmp = 2.0d0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0d0)) * k)
    else
        tmp = 2.0d0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * k) * 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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))))) <= 5e+202) {
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * k) * 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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))) <= 5e+202:
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k)
	else:
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((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)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))))) <= 5e+202)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * Float64(t_m * t_m)) * Float64(k * 2.0)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / Float64(Float64(l / t_m) * l)) * t_m) * Float64(Float64(k * k) * 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))))) <= 5e+202)
		tmp = 2.0 / ((((t_m / (l * l)) * (t_m * t_m)) * (k * 2.0)) * k);
	else
		tmp = 2.0 / (((t_m / ((l / t_m) * l)) * t_m) * ((k * k) * 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[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+202], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.9999999999999999e202

   1. Initial program 77.2%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 75.5%

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

   9. Applied rewrites 74.5%

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

   if 4.9999999999999999e202 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 19.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 32.0

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

   5. Applied rewrites 32.0%

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

   7. Applied rewrites 35.3%

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

   9. Applied rewrites 43.7%

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

   11. Applied rewrites 42.3%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+202}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{t} \cdot \ell} \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5} \cdot k}{\ell}\\ t_3 := 1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(t\_2 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot t\_3}\\ \end{array} \end{array} \end{array} \]

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 t_m 1.5) k) l))
        (t_3 (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 1.35e-71)
      (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (/ (* k k) l) t_m) l)))
      (if (<= t_m 5.6e+163)
        (/ 2.0 (* t_3 (* t_2 t_2)))
        (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l) (/ t_m l)) t_3)))))))

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(t_m, 1.5) * k) / l;
	double t_3 = 1.0 + (1.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 1.35e-71) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	} else if (t_m <= 5.6e+163) {
		tmp = 2.0 / (t_3 * (t_2 * t_2));
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l) * (t_m / l)) * t_3);
	}
	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) :: t_3
    real(8) :: tmp
    t_2 = ((t_m ** 1.5d0) * k) / l
    t_3 = 1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))
    if (t_m <= 1.35d-71) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k * k) / l) * t_m) / l))
    else if (t_m <= 5.6d+163) then
        tmp = 2.0d0 / (t_3 * (t_2 * t_2))
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / l) * (t_m / l)) * t_3)
    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(t_m, 1.5) * k) / l;
	double t_3 = 1.0 + (1.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 1.35e-71) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k * k) / l) * t_m) / l));
	} else if (t_m <= 5.6e+163) {
		tmp = 2.0 / (t_3 * (t_2 * t_2));
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / l) * (t_m / l)) * t_3);
	}
	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(t_m, 1.5) * k) / l
	t_3 = 1.0 + (1.0 + math.pow((k / t_m), 2.0))
	tmp = 0
	if t_m <= 1.35e-71:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k * k) / l) * t_m) / l))
	elif t_m <= 5.6e+163:
		tmp = 2.0 / (t_3 * (t_2 * t_2))
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / l) * (t_m / l)) * t_3)
	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 ^ 1.5) * k) / l)
	t_3 = Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 1.35e-71)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k * k) / l) * t_m) / l)));
	elseif (t_m <= 5.6e+163)
		tmp = Float64(2.0 / Float64(t_3 * Float64(t_2 * t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l) * Float64(t_m / l)) * t_3));
	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 = ((t_m ^ 1.5) * k) / l;
	t_3 = 1.0 + (1.0 + ((k / t_m) ^ 2.0));
	tmp = 0.0;
	if (t_m <= 1.35e-71)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	elseif (t_m <= 5.6e+163)
		tmp = 2.0 / (t_3 * (t_2 * t_2));
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / l) * (t_m / l)) * t_3);
	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[(N[Power[t$95$m, 1.5], $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.35e-71], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+163], N[(2.0 / N[(t$95$3 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3500000000000001e-71

   1. Initial program 46.0%

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

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 68.8%

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

   if 1.3500000000000001e-71 < t < 5.60000000000000029e163

   1. Initial program 54.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 51.6

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

   7. Applied rewrites 51.6%

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

   9. Applied rewrites 77.1%

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

   if 5.60000000000000029e163 < t

   1. Initial program 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 63.6

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

   7. Applied rewrites 63.6%

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

   9. Applied rewrites 85.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t}{\ell}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{{t}^{1.5} \cdot k}{\ell} \cdot \frac{{t}^{1.5} \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.2e-72)
    (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (/ (* k k) l) t_m) l)))
    (/
     2.0
     (*
      (* (* (* (/ t_m l) t_m) (* (+ (pow (/ k t_m) 2.0) 2.0) (sin k))) (tan k))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.2e-72) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * ((pow((k / t_m), 2.0) + 2.0) * sin(k))) * tan(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9.2d-72) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k * k) / l) * t_m) / l))
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) * ((((k / t_m) ** 2.0d0) + 2.0d0) * sin(k))) * tan(k)) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.2e-72) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k * k) / l) * t_m) / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * ((Math.pow((k / t_m), 2.0) + 2.0) * Math.sin(k))) * Math.tan(k)) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9.2e-72:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k * k) / l) * t_m) / l))
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) * ((math.pow((k / t_m), 2.0) + 2.0) * math.sin(k))) * math.tan(k)) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.2e-72)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k * k) / l) * t_m) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * sin(k))) * tan(k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9.2e-72)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	else
		tmp = 2.0 / (((((t_m / l) * t_m) * ((((k / t_m) ^ 2.0) + 2.0) * sin(k))) * tan(k)) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.2e-72], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.19999999999999978e-72

   1. Initial program 46.0%

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

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 68.8%

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

   if 9.19999999999999978e-72 < t

   1. Initial program 57.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 77.4

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

   4. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 80.1

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 80.1%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   10. Applied rewrites 83.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\_m\right)}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (+ 1.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 1.35e-71)
      (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (/ (* (/ (* k k) l) t_m) l)))
      (if (<= t_m 1.05e+44)
        (/ 2.0 (* (* (pow (* (/ t_m l) k) 2.0) t_m) t_2))
        (/ 2.0 (* (* (/ (pow (* k t_m) 2.0) l) (/ t_m l)) 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 + (1.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 1.35e-71) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	} else if (t_m <= 1.05e+44) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * t_2);
	} else {
		tmp = 2.0 / (((pow((k * t_m), 2.0) / l) * (t_m / l)) * 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 + (1.0d0 + ((k / t_m) ** 2.0d0))
    if (t_m <= 1.35d-71) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * ((((k * k) / l) * t_m) / l))
    else if (t_m <= 1.05d+44) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * t_2)
    else
        tmp = 2.0d0 / (((((k * t_m) ** 2.0d0) / l) * (t_m / l)) * 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 + (1.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 1.35e-71) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((((k * k) / l) * t_m) / l));
	} else if (t_m <= 1.05e+44) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * t_2);
	} else {
		tmp = 2.0 / (((Math.pow((k * t_m), 2.0) / l) * (t_m / l)) * 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 + (1.0 + math.pow((k / t_m), 2.0))
	tmp = 0
	if t_m <= 1.35e-71:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((((k * k) / l) * t_m) / l))
	elif t_m <= 1.05e+44:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * t_2)
	else:
		tmp = 2.0 / (((math.pow((k * t_m), 2.0) / l) * (t_m / l)) * 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(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 1.35e-71)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(Float64(Float64(k * k) / l) * t_m) / l)));
	elseif (t_m <= 1.05e+44)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k * t_m) ^ 2.0) / l) * Float64(t_m / l)) * 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 + (1.0 + ((k / t_m) ^ 2.0));
	tmp = 0.0;
	if (t_m <= 1.35e-71)
		tmp = 2.0 / (((sin(k) ^ 2.0) / cos(k)) * ((((k * k) / l) * t_m) / l));
	elseif (t_m <= 1.05e+44)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * t_2);
	else
		tmp = 2.0 / (((((k * t_m) ^ 2.0) / l) * (t_m / l)) * 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[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.35e-71], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+44], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.3500000000000001e-71

   1. Initial program 46.0%

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

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 68.8%

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

   if 1.3500000000000001e-71 < t < 1.04999999999999993e44

   1. Initial program 61.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 65.6

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 58.0

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 57.2%

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

   11. Applied rewrites 74.7%

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

   if 1.04999999999999993e44 < t

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 83.9

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 55.8

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

   7. Applied rewrites 55.8%

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

   9. Applied rewrites 83.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{\frac{k \cdot k}{\ell} \cdot t}{\ell}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 0.18:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \end{array} \]

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 1.7e-65)
    (/
     2.0
     (* (* (pow (* (/ t_m l) k) 2.0) t_m) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
    (if (<= k 0.18)
      (/
       2.0
       (*
        (*
         (fma
          (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
          k
          (* (/ (* t_m t_m) l) 2.0))
         (* k k))
        (/ t_m l)))
      (/ 2.0 (* (* (* k k) t_m) (/ (pow (sin k) 2.0) (* (* (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-65) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else if (k <= 0.18) {
		tmp = 2.0 / ((fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)) * (t_m / l));
	} else {
		tmp = 2.0 / (((k * k) * t_m) * (pow(sin(k), 2.0) / ((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-65)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	elseif (k <= 0.18)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k)) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64((sin(k) ^ 2.0) / 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, 1.7e-65], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.18], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.69999999999999993e-65

   1. Initial program 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 34.0

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 53.0

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 63.4%

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

   11. Applied rewrites 76.5%

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

   if 1.69999999999999993e-65 < k < 0.17999999999999999

   1. Initial program 66.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 26.5

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

   4. Applied rewrites 26.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 26.5

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 26.5%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 99.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)} \cdot \frac{t}{\ell}} \]

   if 0.17999999999999999 < k

   1. Initial program 40.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 27.1

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

   4. Applied rewrites 27.1%

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

   5. Taylor expanded in t around 0

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

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 58.8

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

   7. Applied rewrites 58.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;k \leq 0.18:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\frac{\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m}{\ell} \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 2e+222)
    (/
     2.0
     (* (* (pow (* (/ t_m l) k) 2.0) t_m) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
    (/ 2.0 (* 2.0 (* (* (/ (* (/ (* t_m t_m) l) t_m) l) (sin k)) (tan k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+222) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / (2.0 * ((((((t_m * t_m) / l) * t_m) / l) * sin(k)) * tan(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 ((l * l) <= 2d+222) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    else
        tmp = 2.0d0 / (2.0d0 * ((((((t_m * t_m) / l) * t_m) / l) * sin(k)) * tan(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 ((l * l) <= 2e+222) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / (2.0 * ((((((t_m * t_m) / l) * t_m) / l) * Math.sin(k)) * Math.tan(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 (l * l) <= 2e+222:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	else:
		tmp = 2.0 / (2.0 * ((((((t_m * t_m) / l) * t_m) / l) * math.sin(k)) * math.tan(k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+222)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_m) / l) * sin(k)) * tan(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 ((l * l) <= 2e+222)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	else
		tmp = 2.0 / (2.0 * ((((((t_m * t_m) / l) * t_m) / l) * sin(k)) * tan(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e+222], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000001e222

   1. Initial program 55.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 35.0

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 57.5

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

   7. Applied rewrites 57.5%

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

   9. Applied rewrites 68.0%

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

   11. Applied rewrites 82.5%

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

   if 2.0000000000000001e222 < (*.f64 l l)

   1. Initial program 34.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 45.9

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

   4. Applied rewrites 45.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 61.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\left(\frac{\frac{t \cdot t}{\ell} \cdot t}{\ell} \cdot \sin k\right) \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+201}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t\_m}{\ell} \cdot k\right)}^{2} \cdot t\_m\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= (* l l) 1e+201)
    (/
     2.0
     (* (* (pow (* (/ t_m l) k) 2.0) t_m) (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))))
    (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+201) {
		tmp = 2.0 / ((pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 ((l * l) <= 1d+201) then
        tmp = 2.0d0 / (((((t_m / l) * k) ** 2.0d0) * t_m) * (1.0d0 + (1.0d0 + ((k / t_m) ** 2.0d0))))
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0d0)) * 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 ((l * l) <= 1e+201) {
		tmp = 2.0 / ((Math.pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 (l * l) <= 1e+201:
		tmp = 2.0 / ((math.pow(((t_m / l) * k), 2.0) * t_m) * (1.0 + (1.0 + math.pow((k / t_m), 2.0))))
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+201)
		tmp = Float64(2.0 / Float64(Float64((Float64(Float64(t_m / l) * k) ^ 2.0) * t_m) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * 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 ((l * l) <= 1e+201)
		tmp = 2.0 / (((((t_m / l) * k) ^ 2.0) * t_m) * (1.0 + (1.0 + ((k / t_m) ^ 2.0))));
	else
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e+201], N[(2.0 / N[(N[(N[Power[N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000004e201

   1. Initial program 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 35.3

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

   4. Applied rewrites 35.3%

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

   5. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 59.3

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

   7. Applied rewrites 59.3%

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

   9. Applied rewrites 69.7%

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

   11. Applied rewrites 84.6%

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

   if 1.00000000000000004e201 < (*.f64 l l)

   1. Initial program 33.4%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 41.5

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

   5. Applied rewrites 41.5%

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

   7. Applied rewrites 43.1%

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

   9. Applied rewrites 53.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+201}:\\ \;\;\;\;\frac{2}{\left({\left(\frac{t}{\ell} \cdot k\right)}^{2} \cdot t\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9e-121)
    (/ 2.0 (* (pow (* (/ (pow t_m 1.5) l) k) 2.0) 2.0))
    (/
     2.0
     (*
      (*
       (fma
        (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
        k
        (* (/ (* t_m t_m) l) 2.0))
       (* k k))
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9e-121) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * k), 2.0) * 2.0);
	} else {
		tmp = 2.0 / ((fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9e-121)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * k) ^ 2.0) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k)) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9e-121], N[(2.0 / N[(N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.0000000000000007e-121

   1. Initial program 50.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 33.5%

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

   if 9.0000000000000007e-121 < 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

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 27.1

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

   4. Applied rewrites 27.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 29.6

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 29.6%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.2%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 63.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)} \cdot \frac{t}{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.45 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{{t\_m}^{-3}} \cdot \frac{k \cdot 2}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 2.2e-12)
    (/
     2.0
     (*
      (*
       (fma
        (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
        k
        (* (/ (* t_m t_m) l) 2.0))
       (* k k))
      (/ t_m l)))
    (if (<= t_m 2.45e+89)
      (/ 2.0 (/ (* (/ k (pow t_m -3.0)) (/ (* k 2.0) l)) l))
      (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) 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 <= 2.2e-12) {
		tmp = 2.0 / ((fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)) * (t_m / l));
	} else if (t_m <= 2.45e+89) {
		tmp = 2.0 / (((k / pow(t_m, -3.0)) * ((k * 2.0) / l)) / l);
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 2.2e-12)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k)) * Float64(t_m / l)));
	elseif (t_m <= 2.45e+89)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / (t_m ^ -3.0)) * Float64(Float64(k * 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * k));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-12], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.45e+89], N[(2.0 / N[(N[(N[(k / N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.19999999999999992e-12

   1. Initial program 46.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.2

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

   4. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 20.7

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 20.7%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)} \cdot \frac{t}{\ell}} \]

   if 2.19999999999999992e-12 < t < 2.44999999999999998e89

   1. Initial program 72.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 86.6%

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

   if 2.44999999999999998e89 < t

   1. Initial program 54.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 76.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{{t}^{-3}} \cdot \frac{k \cdot 2}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right)}{\ell} \cdot k, k, \frac{t\_m \cdot t\_m}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m \cdot t\_m}} \cdot \frac{k \cdot 2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 2.2e-12)
    (/
     2.0
     (*
      (*
       (fma
        (* (/ (fma 0.3333333333333333 (* t_m t_m) 1.0) l) k)
        k
        (* (/ (* t_m t_m) l) 2.0))
       (* k k))
      (/ t_m l)))
    (if (<= t_m 1.35e+121)
      (/ 2.0 (* (/ (* k t_m) (/ l (* t_m t_m))) (/ (* k 2.0) l)))
      (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) 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 <= 2.2e-12) {
		tmp = 2.0 / ((fma(((fma(0.3333333333333333, (t_m * t_m), 1.0) / l) * k), k, (((t_m * t_m) / l) * 2.0)) * (k * k)) * (t_m / l));
	} else if (t_m <= 1.35e+121) {
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 2.2e-12)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) / l) * k), k, Float64(Float64(Float64(t_m * t_m) / l) * 2.0)) * Float64(k * k)) * Float64(t_m / l)));
	elseif (t_m <= 1.35e+121)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / Float64(t_m * t_m))) * Float64(Float64(k * 2.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * k));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-12], N[(2.0 / N[(N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+121], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.19999999999999992e-12

   1. Initial program 46.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.2

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

   4. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 20.7

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 20.7%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 64.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right)} \cdot \frac{t}{\ell}} \]

   if 2.19999999999999992e-12 < t < 1.3500000000000001e121

   1. Initial program 72.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 65.6%

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

   9. Applied rewrites 83.2%

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

   if 1.3500000000000001e121 < t

   1. Initial program 53.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 56.6

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 75.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right)}{\ell} \cdot k, k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t \cdot t}} \cdot \frac{k \cdot 2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{t\_m \cdot t\_m} + 0.3333333333333333, k \cdot k, 2\right) \cdot k\right) \cdot k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.2e-107)
    (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m)))))
    (/
     2.0
     (*
      (*
       (*
        (*
         (* (fma (+ (/ 1.0 (* t_m t_m)) 0.3333333333333333) (* k k) 2.0) k)
         k)
        (/ t_m l))
       t_m)
      (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-107) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = 2.0 / (((((fma(((1.0 / (t_m * t_m)) + 0.3333333333333333), (k * k), 2.0) * k) * k) * (t_m / l)) * t_m) * (t_m / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.2e-107)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(1.0 / Float64(t_m * t_m)) + 0.3333333333333333), Float64(k * k), 2.0) * k) * k) * Float64(t_m / l)) * t_m) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.2e-107], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999997e-107

   1. Initial program 50.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 52.2

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

   5. Applied rewrites 52.2%

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

   7. Applied rewrites 62.6%

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

   if 1.19999999999999997e-107 < k

   1. Initial program 44.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 26.5

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

   4. Applied rewrites 26.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 29.1

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 29.1%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 67.3%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

           \[\leadsto \frac{2}{\left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{t \cdot t} + \frac{1}{3}, \color{blue}{k \cdot k}, 2\right) \cdot k\right) \cdot k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       15. lower-*.f64 56.7

           \[\leadsto \frac{2}{\left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{t \cdot t} + 0.3333333333333333, \color{blue}{k \cdot k}, 2\right) \cdot k\right) \cdot k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 56.7%

       \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\left(\mathsf{fma}\left(\frac{1}{t \cdot t} + 0.3333333333333333, k \cdot k, 2\right) \cdot k\right) \cdot k\right)} \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{t \cdot t} + 0.3333333333333333, k \cdot k, 2\right) \cdot k\right) \cdot k\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.92 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t\_m}} \cdot \frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 1.92e-75)
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) t_m) (/ t_m l)))
    (if (<= t_m 1e+121)
      (/ 2.0 (* (/ k (/ l t_m)) (/ (* k 2.0) (/ l (* t_m t_m)))))
      (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) 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 <= 1.92e-75) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else if (t_m <= 1e+121) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.92d-75) then
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * t_m) * (t_m / l))
    else if (t_m <= 1d+121) then
        tmp = 2.0d0 / ((k / (l / t_m)) * ((k * 2.0d0) / (l / (t_m * t_m))))
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0d0)) * 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 <= 1.92e-75) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else if (t_m <= 1e+121) {
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.92e-75:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l))
	elif t_m <= 1e+121:
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))))
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.92e-75)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * t_m) * Float64(t_m / l)));
	elseif (t_m <= 1e+121)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / t_m)) * Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * 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 <= 1.92e-75)
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	elseif (t_m <= 1e+121)
		tmp = 2.0 / ((k / (l / t_m)) * ((k * 2.0) / (l / (t_m * t_m))));
	else
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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, 1.92e-75], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+121], N[(2.0 / N[(N[(k / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.92000000000000011e-75

   1. Initial program 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 16.8

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

   4. Applied rewrites 16.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 18.3

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 18.3%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 61.9

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

   11. Applied rewrites 61.9%

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

   if 1.92000000000000011e-75 < t < 1.00000000000000004e121

   1. Initial program 63.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 71.7%

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

   if 1.00000000000000004e121 < t

   1. Initial program 53.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 56.6

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 75.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.92 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k \cdot 2}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t\_m}{\frac{\ell}{t\_m \cdot t\_m}} \cdot \frac{k \cdot 2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 1.15e-36)
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) t_m) (/ t_m l)))
    (if (<= t_m 1.35e+121)
      (/ 2.0 (* (/ (* k t_m) (/ l (* t_m t_m))) (/ (* k 2.0) l)))
      (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) 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 <= 1.15e-36) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else if (t_m <= 1.35e+121) {
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.15d-36) then
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * t_m) * (t_m / l))
    else if (t_m <= 1.35d+121) then
        tmp = 2.0d0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0d0) / l))
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0d0)) * 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 <= 1.15e-36) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else if (t_m <= 1.35e+121) {
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.15e-36:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l))
	elif t_m <= 1.35e+121:
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l))
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.15e-36)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * t_m) * Float64(t_m / l)));
	elseif (t_m <= 1.35e+121)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) / Float64(l / Float64(t_m * t_m))) * Float64(Float64(k * 2.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * 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 <= 1.15e-36)
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	elseif (t_m <= 1.35e+121)
		tmp = 2.0 / (((k * t_m) / (l / (t_m * t_m))) * ((k * 2.0) / l));
	else
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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, 1.15e-36], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+121], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * 2.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.14999999999999998e-36

   1. Initial program 45.9%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 18.2

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

   4. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 19.7

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 19.7%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.0%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 61.4

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

   11. Applied rewrites 61.4%

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

   if 1.14999999999999998e-36 < t < 1.3500000000000001e121

   1. Initial program 69.3%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 60.1

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

   5. Applied rewrites 60.1%

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

   7. Applied rewrites 60.1%

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

   9. Applied rewrites 74.4%

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

   if 1.3500000000000001e121 < t

   1. Initial program 53.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 56.6

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 75.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-36}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\frac{\ell}{t \cdot t}} \cdot \frac{k \cdot 2}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.92 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 10^{+121}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot 2}{\frac{\ell}{t\_m \cdot t\_m}} \cdot \frac{t\_m}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 1.92e-75)
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) t_m) (/ t_m l)))
    (if (<= t_m 1e+121)
      (/ 2.0 (* (* (/ (* k 2.0) (/ l (* t_m t_m))) (/ t_m l)) k))
      (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) 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 <= 1.92e-75) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else if (t_m <= 1e+121) {
		tmp = 2.0 / ((((k * 2.0) / (l / (t_m * t_m))) * (t_m / l)) * k);
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.92d-75) then
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * t_m) * (t_m / l))
    else if (t_m <= 1d+121) then
        tmp = 2.0d0 / ((((k * 2.0d0) / (l / (t_m * t_m))) * (t_m / l)) * k)
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0d0)) * 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 <= 1.92e-75) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else if (t_m <= 1e+121) {
		tmp = 2.0 / ((((k * 2.0) / (l / (t_m * t_m))) * (t_m / l)) * k);
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.92e-75:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l))
	elif t_m <= 1e+121:
		tmp = 2.0 / ((((k * 2.0) / (l / (t_m * t_m))) * (t_m / l)) * k)
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 1.92e-75)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * t_m) * Float64(t_m / l)));
	elseif (t_m <= 1e+121)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * 2.0) / Float64(l / Float64(t_m * t_m))) * Float64(t_m / l)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * 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 <= 1.92e-75)
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	elseif (t_m <= 1e+121)
		tmp = 2.0 / ((((k * 2.0) / (l / (t_m * t_m))) * (t_m / l)) * k);
	else
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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, 1.92e-75], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+121], N[(2.0 / N[(N[(N[(N[(k * 2.0), $MachinePrecision] / N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.92000000000000011e-75

   1. Initial program 46.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 16.8

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

   4. Applied rewrites 16.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 18.3

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 18.3%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       4. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       7. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       8. lower-*.f64 61.9

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 61.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \frac{k \cdot k}{\ell}\right) \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

   if 1.92000000000000011e-75 < t < 1.00000000000000004e121

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 56.9

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 56.9%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot \color{blue}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.6%

      \[\leadsto \frac{2}{\left(\frac{k \cdot 2}{\frac{\ell}{t \cdot t}} \cdot \frac{t}{\ell}\right) \cdot k} \]

   if 1.00000000000000004e121 < t

   1. Initial program 53.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 56.6

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 56.6%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.6%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot \color{blue}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.0%

      \[\leadsto \frac{2}{\left(\left(\frac{-t}{\ell} \cdot \left(\left(-t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.92 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 10^{+121}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot 2}{\frac{\ell}{t \cdot t}} \cdot \frac{t}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot 2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot 2\right) \cdot \frac{\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m}{\ell}\right) \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot t\_m\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* (* k k) 2.0)))
   (*
    t_s
    (if (<= k 1e-158)
      (/ 2.0 (* (* (* k 2.0) (/ (* (/ (* t_m t_m) l) t_m) l)) k))
      (if (<= k 2.1e+41)
        (/ 2.0 (* (* (* (/ t_m l) t_m) (/ t_m l)) t_2))
        (/ 2.0 (* (* (* (/ t_m (* l l)) t_m) t_m) 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 = (k * k) * 2.0;
	double tmp;
	if (k <= 1e-158) {
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	} else if (k <= 2.1e+41) {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0d0
    if (k <= 1d-158) then
        tmp = 2.0d0 / (((k * 2.0d0) * ((((t_m * t_m) / l) * t_m) / l)) * k)
    else if (k <= 2.1d+41) then
        tmp = 2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2)
    else
        tmp = 2.0d0 / ((((t_m / (l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0;
	double tmp;
	if (k <= 1e-158) {
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	} else if (k <= 2.1e+41) {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0
	tmp = 0
	if k <= 1e-158:
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k)
	elif k <= 2.1e+41:
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2)
	else:
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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(k * k) * 2.0)
	tmp = 0.0
	if (k <= 1e-158)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * 2.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_m) / l)) * k));
	elseif (k <= 2.1e+41)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0;
	tmp = 0.0;
	if (k <= 1e-158)
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	elseif (k <= 2.1e+41)
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	else
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1e-158], N[(2.0 / N[(N[(N[(k * 2.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.1e+41], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.00000000000000006e-158

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 50.8

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot \color{blue}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 56.8%

      \[\leadsto \frac{2}{\left(\frac{\frac{t \cdot t}{\ell} \cdot t}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot k} \]

   if 1.00000000000000006e-158 < k < 2.1e41

   1. Initial program 53.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 75.6

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 75.6%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.4%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.7%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \frac{1}{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.0%

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]

   if 2.1e41 < k

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 36.1

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 36.1%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.8%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 45.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-158}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot 2\right) \cdot \frac{\frac{t \cdot t}{\ell} \cdot t}{\ell}\right) \cdot k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 7e-26)
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) t_m) (/ t_m l)))
    (/ 2.0 (* (* (* (* (/ t_m l) t_m) (/ t_m l)) (* k 2.0)) 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 <= 7e-26) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 7d-26) then
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * t_m) * (t_m / l))
    else
        tmp = 2.0d0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0d0)) * 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 <= 7e-26) {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	} else {
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 7e-26:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l))
	else:
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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 <= 7e-26)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * t_m) * Float64(t_m / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * Float64(k * 2.0)) * 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 <= 7e-26)
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	else
		tmp = 2.0 / (((((t_m / l) * t_m) * (t_m / l)) * (k * 2.0)) * 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, 7e-26], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.9999999999999997e-26

   1. Initial program 45.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 18.4

          \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 18.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{2}}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{{t}^{\frac{3}{2}}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 19.9

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 19.9%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 67.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       4. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       7. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       8. lower-*.f64 61.0

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 61.0%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \frac{k \cdot k}{\ell}\right) \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

   if 6.9999999999999997e-26 < t

   1. Initial program 60.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 59.3

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 59.3%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot \color{blue}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.9%

      \[\leadsto \frac{2}{\left(\left(\frac{-t}{\ell} \cdot \left(\left(-t\right) \cdot \frac{t}{\ell}\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot 2\right) \cdot \frac{\frac{t\_m \cdot t\_m}{\ell} \cdot t\_m}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\_m\right) \cdot 2\right) \cdot t\_m\right) \cdot \frac{t\_m}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.15e-120)
    (/ 2.0 (* (* (* k 2.0) (/ (* (/ (* t_m t_m) l) t_m) l)) k))
    (/ 2.0 (* (* (* (* (/ (* k k) l) t_m) 2.0) t_m) (/ t_m l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-120) {
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	} else {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.15d-120) then
        tmp = 2.0d0 / (((k * 2.0d0) * ((((t_m * t_m) / l) * t_m) / l)) * k)
    else
        tmp = 2.0d0 / ((((((k * k) / l) * t_m) * 2.0d0) * t_m) * (t_m / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-120) {
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	} else {
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.15e-120:
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k)
	else:
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.15e-120)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * 2.0) * Float64(Float64(Float64(Float64(t_m * t_m) / l) * t_m) / l)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k * k) / l) * t_m) * 2.0) * t_m) * Float64(t_m / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.15e-120)
		tmp = 2.0 / (((k * 2.0) * ((((t_m * t_m) / l) * t_m) / l)) * k);
	else
		tmp = 2.0 / ((((((k * k) / l) * t_m) * 2.0) * t_m) * (t_m / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.15e-120], N[(2.0 / N[(N[(N[(k * 2.0), $MachinePrecision] * N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.14999999999999993e-120

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 51.6

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 51.6%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.9%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot \color{blue}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

      \[\leadsto \frac{2}{\left(\frac{\frac{t \cdot t}{\ell} \cdot t}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot k} \]

   if 1.14999999999999993e-120 < 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

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 27.1

          \[\leadsto \frac{2}{\left(\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 27.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left(\frac{{t}^{\frac{3}{2}}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{{t}^{\frac{3}{2}}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\left({\left(\frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot {t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{\left(\frac{\frac{3}{2}}{2}\right)} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left(\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\color{blue}{\frac{3}{4}}} \cdot \frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \color{blue}{\frac{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left({\left({t}^{\frac{3}{4}} \cdot \frac{\color{blue}{{t}^{\left(\frac{\frac{3}{2}}{2}\right)}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. metadata-eval 29.6

          \[\leadsto \frac{2}{\left(\left({\left({t}^{0.75} \cdot \frac{{t}^{\color{blue}{0.75}}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 29.6%

      \[\leadsto \frac{2}{\left(\left({\color{blue}{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}}^{2} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   8. Applied rewrites 68.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\tan k \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t\right) \cdot \frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(2 \cdot \frac{{k}^{2} \cdot t}{\ell}\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{k}^{2} \cdot t}{\ell} \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]

       3. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot {k}^{2}}}{\ell} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       4. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)} \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       6. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\frac{{k}^{2}}{\ell}}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       7. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

       8. lower-*.f64 59.5

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 59.5%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(t \cdot \frac{k \cdot k}{\ell}\right) \cdot 2\right)} \cdot t\right) \cdot \frac{t}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot 2\right) \cdot \frac{\frac{t \cdot t}{\ell} \cdot t}{\ell}\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{k \cdot k}{\ell} \cdot t\right) \cdot 2\right) \cdot t\right) \cdot \frac{t}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 60.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot 2\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot t\_m\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* (* k k) 2.0)))
   (*
    t_s
    (if (<= k 2.1e+41)
      (/ 2.0 (* (* (* (/ t_m l) t_m) (/ t_m l)) t_2))
      (/ 2.0 (* (* (* (/ t_m (* l l)) t_m) t_m) 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 = (k * k) * 2.0;
	double tmp;
	if (k <= 2.1e+41) {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0d0
    if (k <= 2.1d+41) then
        tmp = 2.0d0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2)
    else
        tmp = 2.0d0 / ((((t_m / (l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0;
	double tmp;
	if (k <= 2.1e+41) {
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	} else {
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0
	tmp = 0
	if k <= 2.1e+41:
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2)
	else:
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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(k * k) * 2.0)
	tmp = 0.0
	if (k <= 2.1e+41)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(t_m / l)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * t_m) * 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 = (k * k) * 2.0;
	tmp = 0.0;
	if (k <= 2.1e+41)
		tmp = 2.0 / ((((t_m / l) * t_m) * (t_m / l)) * t_2);
	else
		tmp = 2.0 / ((((t_m / (l * l)) * t_m) * t_m) * 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[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.1e+41], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e41

   1. Initial program 51.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 55.0

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 55.0%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.0%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \frac{1}{\color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}\right)} \]

   if 2.1e41 < k

   1. Initial program 37.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 36.1

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 36.1%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.8%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 45.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 58.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell \cdot \ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot t\_m\right) \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ t_m (* l l))))
   (*
    t_s
    (if (<= k 8.5e-156)
      (/ 2.0 (* (* (* t_2 (* t_m t_m)) (* k 2.0)) k))
      (/ 2.0 (* (* (* t_2 t_m) t_m) (* (* 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 / (l * l);
	double tmp;
	if (k <= 8.5e-156) {
		tmp = 2.0 / (((t_2 * (t_m * t_m)) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_2 * t_m) * t_m) * ((k * k) * 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) :: t_2
    real(8) :: tmp
    t_2 = t_m / (l * l)
    if (k <= 8.5d-156) then
        tmp = 2.0d0 / (((t_2 * (t_m * t_m)) * (k * 2.0d0)) * k)
    else
        tmp = 2.0d0 / (((t_2 * t_m) * t_m) * ((k * k) * 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 t_2 = t_m / (l * l);
	double tmp;
	if (k <= 8.5e-156) {
		tmp = 2.0 / (((t_2 * (t_m * t_m)) * (k * 2.0)) * k);
	} else {
		tmp = 2.0 / (((t_2 * t_m) * t_m) * ((k * k) * 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):
	t_2 = t_m / (l * l)
	tmp = 0
	if k <= 8.5e-156:
		tmp = 2.0 / (((t_2 * (t_m * t_m)) * (k * 2.0)) * k)
	else:
		tmp = 2.0 / (((t_2 * t_m) * t_m) * ((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(t_m / Float64(l * l))
	tmp = 0.0
	if (k <= 8.5e-156)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(t_m * t_m)) * Float64(k * 2.0)) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * t_m) * t_m) * Float64(Float64(k * k) * 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)
	t_2 = t_m / (l * l);
	tmp = 0.0;
	if (k <= 8.5e-156)
		tmp = 2.0 / (((t_2 * (t_m * t_m)) * (k * 2.0)) * k);
	else
		tmp = 2.0 / (((t_2 * t_m) * t_m) * ((k * k) * 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_] := Block[{t$95$2 = N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 8.5e-156], N[(2.0 / N[(N[(N[(t$95$2 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.5e-156

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 50.8

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 50.8%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(k \cdot 2\right)\right) \cdot \color{blue}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.8%

      \[\leadsto \frac{2}{\left(\left(\left(t \cdot t\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot 2\right)\right) \cdot k} \]

   if 8.5e-156 < k

   1. Initial program 44.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

      11. lower-pow.f64 54.1

          \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

   5. Applied rewrites 54.1%

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.1%

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot \left(t \cdot t\right)\right) \cdot \left(k \cdot 2\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 57.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(\left(\frac{t\_m}{\ell \cdot \ell} \cdot t\_m\right) \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \end{array} \]

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 (/ 2.0 (* (* (* (/ t_m (* l l)) t_m) t_m) (* (* 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) {
	return t_s * (2.0 / ((((t_m / (l * l)) * t_m) * t_m) * ((k * k) * 2.0)));
}

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 * (2.0d0 / ((((t_m / (l * l)) * t_m) * t_m) * ((k * k) * 2.0d0)))
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 * (2.0 / ((((t_m / (l * l)) * t_m) * t_m) * ((k * k) * 2.0)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((((t_m / (l * l)) * t_m) * t_m) * ((k * k) * 2.0)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(Float64(t_m / Float64(l * l)) * t_m) * t_m) * Float64(Float64(k * k) * 2.0))))
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 * (2.0 / ((((t_m / (l * l)) * t_m) * t_m) * ((k * k) * 2.0)));
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[(2.0 / N[(N[(N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \frac{2}{2 \cdot \color{blue}{\left({k}^{2} \cdot \frac{{t}^{3}}{{\ell}^{2}}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {k}^{2}\right)} \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{t}^{3}}{{\ell}^{2}}} \]

   7. unpow2 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\color{blue}{\frac{{t}^{3}}{\ell}}}{\ell}} \]

   11. lower-pow.f64 51.8

       \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{\color{blue}{{t}^{3}}}{\ell}}{\ell}} \]

5. Applied rewrites 51.8%

   \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 52.7%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{t \cdot t}{\ell}}{\ell}}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 53.6%

   \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \left(t \cdot \color{blue}{\frac{t}{\ell \cdot \ell}}\right)\right)} \]

10. Final simplification 53.6%

    \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell \cdot \ell} \cdot t\right) \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot 2\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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))))