Toniolo and Linder, Equation (10+)

Percentage Accurate: 56.1% → 95.2%

Time: 12.8s

Alternatives: 23

Speedup: 10.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-87)
    (/ (* (* (/ 2.0 k) (/ l (* k t_m))) (* (cos k) l)) (pow (sin k) 2.0))
    (/
     (/ 2.0 (/ (* t_m (sin k)) l))
     (* (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (/ t_m l)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else {
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.85d-87) then
        tmp = (((2.0d0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ** 2.0d0)
    else
        tmp = (2.0d0 / ((t_m * sin(k)) / l)) / ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * (t_m / l)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (Math.cos(k) * l)) / Math.pow(Math.sin(k), 2.0);
	} else {
		tmp = (2.0 / ((t_m * Math.sin(k)) / l)) / ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.85e-87:
		tmp = (((2.0 / k) * (l / (k * t_m))) * (math.cos(k) * l)) / math.pow(math.sin(k), 2.0)
	else:
		tmp = (2.0 / ((t_m * math.sin(k)) / l)) / ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * (t_m / l)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.85e-87)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * sin(k)) / l)) / Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.85e-87)
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ^ 2.0);
	else
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * (t_m / l)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-87], N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-87

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 77.5%

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

   if 1.8500000000000001e-87 < t

   1. Initial program 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 93.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 93.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 94.7%

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

Alternative 2: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \frac{\ell}{k}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       4e+294)
    (/ (* (/ l (* t_m t_m)) (/ l k)) (* t_m k))
    (* (/ 2.0 (* (* k k) t_m)) (* l (/ l (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 4e+294) {
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k);
	} else {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 4d+294) then
        tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k)
    else
        tmp = (2.0d0 / ((k * k) * t_m)) * (l * (l / (k * k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+294) {
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k);
	} else {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 4e+294:
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k)
	else:
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 4e+294)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * Float64(l / k)) / Float64(t_m * k));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(l * Float64(l / Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 4e+294)
		tmp = ((l / (t_m * t_m)) * (l / k)) / (t_m * k);
	else
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(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[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+294], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.00000000000000027e294

   1. Initial program 84.0%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 82.2%

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

   if 4.00000000000000027e294 < (/.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 24.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 52.6%

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

Alternative 3: 93.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \left(\tan k \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right)\right) \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_2}{\ell}}}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= t_m 4.2e-48)
      (/ (* (* (/ 2.0 k) (/ l (* k t_m))) (* (cos k) l)) (pow (sin k) 2.0))
      (if (<= t_m 3.2e+157)
        (*
         (/
          2.0
          (*
           (* t_2 (* (tan k) (* (/ t_m l) t_m)))
           (+ (pow (/ k t_m) 2.0) 2.0)))
         l)
        (/ (/ 2.0 (/ t_2 l)) (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * sin(k);
	double tmp;
	if (t_m <= 4.2e-48) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else if (t_m <= 3.2e+157) {
		tmp = (2.0 / ((t_2 * (tan(k) * ((t_m / l) * t_m))) * (pow((k / t_m), 2.0) + 2.0))) * l;
	} else {
		tmp = (2.0 / (t_2 / l)) / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sin(k)
    if (t_m <= 4.2d-48) then
        tmp = (((2.0d0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ** 2.0d0)
    else if (t_m <= 3.2d+157) then
        tmp = (2.0d0 / ((t_2 * (tan(k) * ((t_m / l) * t_m))) * (((k / t_m) ** 2.0d0) + 2.0d0))) * l
    else
        tmp = (2.0d0 / (t_2 / l)) / (((2.0d0 * tan(k)) * (t_m / l)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * Math.sin(k);
	double tmp;
	if (t_m <= 4.2e-48) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (Math.cos(k) * l)) / Math.pow(Math.sin(k), 2.0);
	} else if (t_m <= 3.2e+157) {
		tmp = (2.0 / ((t_2 * (Math.tan(k) * ((t_m / l) * t_m))) * (Math.pow((k / t_m), 2.0) + 2.0))) * l;
	} else {
		tmp = (2.0 / (t_2 / l)) / (((2.0 * Math.tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * math.sin(k)
	tmp = 0
	if t_m <= 4.2e-48:
		tmp = (((2.0 / k) * (l / (k * t_m))) * (math.cos(k) * l)) / math.pow(math.sin(k), 2.0)
	elif t_m <= 3.2e+157:
		tmp = (2.0 / ((t_2 * (math.tan(k) * ((t_m / l) * t_m))) * (math.pow((k / t_m), 2.0) + 2.0))) * l
	else:
		tmp = (2.0 / (t_2 / l)) / (((2.0 * math.tan(k)) * (t_m / l)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (t_m <= 4.2e-48)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	elseif (t_m <= 3.2e+157)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_2 * Float64(tan(k) * Float64(Float64(t_m / l) * t_m))) * Float64((Float64(k / t_m) ^ 2.0) + 2.0))) * l);
	else
		tmp = Float64(Float64(2.0 / Float64(t_2 / l)) / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * sin(k);
	tmp = 0.0;
	if (t_m <= 4.2e-48)
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ^ 2.0);
	elseif (t_m <= 3.2e+157)
		tmp = (2.0 / ((t_2 * (tan(k) * ((t_m / l) * t_m))) * (((k / t_m) ^ 2.0) + 2.0))) * l;
	else
		tmp = (2.0 / (t_2 / l)) / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-48], N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.2e+157], N[(N[(2.0 / N[(N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(2.0 / N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 4.19999999999999977e-48

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.4

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

   5. Applied rewrites 62.4%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 77.4%

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

   if 4.19999999999999977e-48 < t < 3.1999999999999999e157

   1. Initial program 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 88.6%

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

   if 3.1999999999999999e157 < t

   1. Initial program 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 96.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 96.6%

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

Alternative 4: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-87)
    (/ (* (* (/ 2.0 k) (/ l (* k t_m))) (* (cos k) l)) (pow (sin k) 2.0))
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (* (* (/ t_m l) t_m) (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else {
		tmp = 2.0 / (((sin(k) * t_m) / l) * (((t_m / l) * t_m) * (tan(k) * (pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.85d-87) then
        tmp = (((2.0d0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ** 2.0d0)
    else
        tmp = 2.0d0 / (((sin(k) * t_m) / l) * (((t_m / l) * t_m) * (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (Math.cos(k) * l)) / Math.pow(Math.sin(k), 2.0);
	} else {
		tmp = 2.0 / (((Math.sin(k) * t_m) / l) * (((t_m / l) * t_m) * (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.85e-87:
		tmp = (((2.0 / k) * (l / (k * t_m))) * (math.cos(k) * l)) / math.pow(math.sin(k), 2.0)
	else:
		tmp = 2.0 / (((math.sin(k) * t_m) / l) * (((t_m / l) * t_m) * (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.85e-87)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(Float64(t_m / l) * t_m) * Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.85e-87)
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ^ 2.0);
	else
		tmp = 2.0 / (((sin(k) * t_m) / l) * (((t_m / l) * t_m) * (tan(k) * (((k / t_m) ^ 2.0) + 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-87], N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-87

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 77.5%

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

   if 1.8500000000000001e-87 < t

   1. Initial program 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 93.6%

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

Alternative 5: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_m \cdot \left(\frac{\sin k}{\ell} \cdot \left(\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e-87)
    (/ (* (* (/ 2.0 k) (/ l (* k t_m))) (* (cos k) l)) (pow (sin k) 2.0))
    (/
     2.0
     (*
      t_m
      (*
       (/ (sin k) l)
       (* (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else {
		tmp = 2.0 / (t_m * ((sin(k) / l) * ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * (t_m / l)) * t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.85d-87) then
        tmp = (((2.0d0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ** 2.0d0)
    else
        tmp = 2.0d0 / (t_m * ((sin(k) / l) * ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * (t_m / l)) * t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (Math.cos(k) * l)) / Math.pow(Math.sin(k), 2.0);
	} else {
		tmp = 2.0 / (t_m * ((Math.sin(k) / l) * ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * (t_m / l)) * t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.85e-87:
		tmp = (((2.0 / k) * (l / (k * t_m))) * (math.cos(k) * l)) / math.pow(math.sin(k), 2.0)
	else:
		tmp = 2.0 / (t_m * ((math.sin(k) / l) * ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * (t_m / l)) * t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.85e-87)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_m * Float64(Float64(sin(k) / l) * Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(t_m / l)) * t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.85e-87)
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / (sin(k) ^ 2.0);
	else
		tmp = 2.0 / (t_m * ((sin(k) / l) * ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * (t_m / l)) * t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e-87], N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$m * N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.8500000000000001e-87

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 77.5%

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

   if 1.8500000000000001e-87 < t

   1. Initial program 70.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 92.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 93.2%

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

Alternative 6: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\frac{t\_m \cdot \sin k}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(\frac{2}{k} \cdot \frac{\ell}{k \cdot t\_m}\right) \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_2}{\left(\left(\frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)}{t\_m \cdot t\_m} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (/ (* t_m (sin k)) l))))
   (*
    t_s
    (if (<= t_m 1.85e-87)
      (/ (* (* (/ 2.0 k) (/ l (* k t_m))) (* (cos k) l)) (pow (sin k) 2.0))
      (if (<= t_m 1.5e+137)
        (/
         t_2
         (*
          (*
           (* (/ (fma (* t_m t_m) 2.0 (* k k)) (* t_m t_m)) (tan k))
           (/ t_m l))
          t_m))
        (/ t_2 (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / ((t_m * sin(k)) / l);
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = (((2.0 / k) * (l / (k * t_m))) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else if (t_m <= 1.5e+137) {
		tmp = t_2 / ((((fma((t_m * t_m), 2.0, (k * k)) / (t_m * t_m)) * tan(k)) * (t_m / l)) * t_m);
	} else {
		tmp = t_2 / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(t_m * sin(k)) / l))
	tmp = 0.0
	if (t_m <= 1.85e-87)
		tmp = Float64(Float64(Float64(Float64(2.0 / k) * Float64(l / Float64(k * t_m))) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	elseif (t_m <= 1.5e+137)
		tmp = Float64(t_2 / Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) / Float64(t_m * t_m)) * tan(k)) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(t_2 / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.85e-87], N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+137], N[(t$95$2 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8500000000000001e-87

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 77.5%

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

   if 1.8500000000000001e-87 < t < 1.5e137

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 80.6

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

   11. Applied rewrites 80.6%

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

   if 1.5e137 < t

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.3%

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

Alternative 7: 91.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\frac{t\_m \cdot \sin k}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{-2 \cdot \ell}{\left(\left(-t\_m\right) \cdot k\right) \cdot k} \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_2}{\left(\left(\frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)}{t\_m \cdot t\_m} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (/ (* t_m (sin k)) l))))
   (*
    t_s
    (if (<= t_m 2.8e-88)
      (/
       (* (/ (* -2.0 l) (* (* (- t_m) k) k)) (* (cos k) l))
       (pow (sin k) 2.0))
      (if (<= t_m 1.5e+137)
        (/
         t_2
         (*
          (*
           (* (/ (fma (* t_m t_m) 2.0 (* k k)) (* t_m t_m)) (tan k))
           (/ t_m l))
          t_m))
        (/ t_2 (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / ((t_m * sin(k)) / l);
	double tmp;
	if (t_m <= 2.8e-88) {
		tmp = (((-2.0 * l) / ((-t_m * k) * k)) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else if (t_m <= 1.5e+137) {
		tmp = t_2 / ((((fma((t_m * t_m), 2.0, (k * k)) / (t_m * t_m)) * tan(k)) * (t_m / l)) * t_m);
	} else {
		tmp = t_2 / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(t_m * sin(k)) / l))
	tmp = 0.0
	if (t_m <= 2.8e-88)
		tmp = Float64(Float64(Float64(Float64(-2.0 * l) / Float64(Float64(Float64(-t_m) * k) * k)) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	elseif (t_m <= 1.5e+137)
		tmp = Float64(t_2 / Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) / Float64(t_m * t_m)) * tan(k)) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(t_2 / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.8e-88], N[(N[(N[(N[(-2.0 * l), $MachinePrecision] / N[(N[((-t$95$m) * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+137], N[(t$95$2 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.79999999999999976e-88

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 74.1%

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

   if 2.79999999999999976e-88 < t < 1.5e137

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 80.6

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

   11. Applied rewrites 80.6%

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

   if 1.5e137 < t

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.3%

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

Alternative 8: 91.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\frac{t\_m \cdot \sin k}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-88}:\\ \;\;\;\;\frac{\left(\frac{2}{\left(k \cdot t\_m\right) \cdot k} \cdot \ell\right) \cdot \left(\cos k \cdot \ell\right)}{{\sin k}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_2}{\left(\left(\frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)}{t\_m \cdot t\_m} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (/ (* t_m (sin k)) l))))
   (*
    t_s
    (if (<= t_m 1e-88)
      (/ (* (* (/ 2.0 (* (* k t_m) k)) l) (* (cos k) l)) (pow (sin k) 2.0))
      (if (<= t_m 1.5e+137)
        (/
         t_2
         (*
          (*
           (* (/ (fma (* t_m t_m) 2.0 (* k k)) (* t_m t_m)) (tan k))
           (/ t_m l))
          t_m))
        (/ t_2 (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / ((t_m * sin(k)) / l);
	double tmp;
	if (t_m <= 1e-88) {
		tmp = (((2.0 / ((k * t_m) * k)) * l) * (cos(k) * l)) / pow(sin(k), 2.0);
	} else if (t_m <= 1.5e+137) {
		tmp = t_2 / ((((fma((t_m * t_m), 2.0, (k * k)) / (t_m * t_m)) * tan(k)) * (t_m / l)) * t_m);
	} else {
		tmp = t_2 / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(t_m * sin(k)) / l))
	tmp = 0.0
	if (t_m <= 1e-88)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(Float64(k * t_m) * k)) * l) * Float64(cos(k) * l)) / (sin(k) ^ 2.0));
	elseif (t_m <= 1.5e+137)
		tmp = Float64(t_2 / Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) / Float64(t_m * t_m)) * tan(k)) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(t_2 / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1e-88], N[(N[(N[(N[(2.0 / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+137], N[(t$95$2 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 9.99999999999999934e-89

   1. Initial program 52.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 63.1

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

   5. Applied rewrites 63.1%

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

   7. Applied rewrites 68.3%

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

   9. Applied rewrites 74.4%

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

   if 9.99999999999999934e-89 < t < 1.5e137

   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

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 71.6

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

   4. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 88.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.9

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 88.9%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 79.0

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

   11. Applied rewrites 79.0%

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

   if 1.5e137 < t

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.3%

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

Alternative 9: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\frac{t\_m \cdot \sin k}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{-87}:\\ \;\;\;\;\left(\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\cos k \cdot \ell\right)\right) \cdot \frac{\ell}{{\sin k}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_2}{\left(\left(\frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)}{t\_m \cdot t\_m} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (/ (* t_m (sin k)) l))))
   (*
    t_s
    (if (<= t_m 1.85e-87)
      (* (* (/ 2.0 (* (* k k) t_m)) (* (cos k) l)) (/ l (pow (sin k) 2.0)))
      (if (<= t_m 1.5e+137)
        (/
         t_2
         (*
          (*
           (* (/ (fma (* t_m t_m) 2.0 (* k k)) (* t_m t_m)) (tan k))
           (/ t_m l))
          t_m))
        (/ t_2 (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / ((t_m * sin(k)) / l);
	double tmp;
	if (t_m <= 1.85e-87) {
		tmp = ((2.0 / ((k * k) * t_m)) * (cos(k) * l)) * (l / pow(sin(k), 2.0));
	} else if (t_m <= 1.5e+137) {
		tmp = t_2 / ((((fma((t_m * t_m), 2.0, (k * k)) / (t_m * t_m)) * tan(k)) * (t_m / l)) * t_m);
	} else {
		tmp = t_2 / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(t_m * sin(k)) / l))
	tmp = 0.0
	if (t_m <= 1.85e-87)
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(cos(k) * l)) * Float64(l / (sin(k) ^ 2.0)));
	elseif (t_m <= 1.5e+137)
		tmp = Float64(t_2 / Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) / Float64(t_m * t_m)) * tan(k)) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(t_2 / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.85e-87], N[(N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+137], N[(t$95$2 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8500000000000001e-87

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 66.9%

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

   if 1.8500000000000001e-87 < t < 1.5e137

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 80.6

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

   11. Applied rewrites 80.6%

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

   if 1.5e137 < t

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 97.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 97.3%

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

Alternative 10: 87.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\frac{t\_m \cdot \sin k}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_2}{\left(\left(\frac{\mathsf{fma}\left(t\_m \cdot t\_m, 2, k \cdot k\right)}{t\_m \cdot t\_m} \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (/ (* t_m (sin k)) l))))
   (*
    t_s
    (if (<= t_m 1.55e-87)
      (/ 2.0 (* (* (/ (* k k) l) (/ (* (sin k) t_m) l)) (tan k)))
      (if (<= t_m 1.5e+137)
        (/
         t_2
         (*
          (*
           (* (/ (fma (* t_m t_m) 2.0 (* k k)) (* t_m t_m)) (tan k))
           (/ t_m l))
          t_m))
        (/ t_2 (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / ((t_m * sin(k)) / l);
	double tmp;
	if (t_m <= 1.55e-87) {
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	} else if (t_m <= 1.5e+137) {
		tmp = t_2 / ((((fma((t_m * t_m), 2.0, (k * k)) / (t_m * t_m)) * tan(k)) * (t_m / l)) * t_m);
	} else {
		tmp = t_2 / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(2.0 / Float64(Float64(t_m * sin(k)) / l))
	tmp = 0.0
	if (t_m <= 1.55e-87)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(Float64(sin(k) * t_m) / l)) * tan(k)));
	elseif (t_m <= 1.5e+137)
		tmp = Float64(t_2 / Float64(Float64(Float64(Float64(fma(Float64(t_m * t_m), 2.0, Float64(k * k)) / Float64(t_m * t_m)) * tan(k)) * Float64(t_m / l)) * t_m));
	else
		tmp = Float64(t_2 / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-87], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+137], N[(t$95$2 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0 + N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.54999999999999999e-87

   1. Initial program 51.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow3 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 64.4

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

   4. Applied rewrites 64.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 59.0

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

   7. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 65.9%

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

   10. Taylor expanded in t around 0

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

       1. unpow2 N/A

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

       2. times-frac N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 66.5

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

   12. Applied rewrites 66.5%

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

   if 1.54999999999999999e-87 < t < 1.5e137

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 73.1

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

   4. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

   6. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\frac{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)}{\color{blue}{t \cdot t}} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

       9. lower-*.f64 80.6

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\frac{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)}{\color{blue}{t \cdot t}} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

   11. Applied rewrites 80.6%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{\frac{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right)}{t \cdot t}} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

   if 1.5e137 < t

   1. Initial program 83.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 99.9

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 97.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. lower-/.f64 97.3

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell}}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\sin k \cdot t}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      8. lower-*.f64 97.3

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{2} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.3%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{2} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\left(\left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m\right)}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \sin k}{\ell}}}{\left(\left(\left(\frac{k}{t\_m} \cdot \frac{k}{t\_m}\right) \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.35e-18)
    (/
     2.0
     (*
      (* (/ t_m l) k)
      (* (* (* (+ (pow (/ k t_m) 2.0) 2.0) (tan k)) (/ t_m l)) t_m)))
    (if (<= k 7.8e+148)
      (/ 2.0 (* (* (/ (* k k) l) (/ (* (sin k) t_m) l)) (tan k)))
      (/
       (/ 2.0 (/ (* t_m (sin k)) l))
       (* (* (* (* (/ k t_m) (/ k t_m)) (tan k)) (/ t_m l)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-18) {
		tmp = 2.0 / (((t_m / l) * k) * ((((pow((k / t_m), 2.0) + 2.0) * tan(k)) * (t_m / l)) * t_m));
	} else if (k <= 7.8e+148) {
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	} else {
		tmp = (2.0 / ((t_m * sin(k)) / l)) / (((((k / t_m) * (k / t_m)) * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.35d-18) then
        tmp = 2.0d0 / (((t_m / l) * k) * ((((((k / t_m) ** 2.0d0) + 2.0d0) * tan(k)) * (t_m / l)) * t_m))
    else if (k <= 7.8d+148) then
        tmp = 2.0d0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k))
    else
        tmp = (2.0d0 / ((t_m * sin(k)) / l)) / (((((k / t_m) * (k / t_m)) * tan(k)) * (t_m / l)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.35e-18) {
		tmp = 2.0 / (((t_m / l) * k) * ((((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(k)) * (t_m / l)) * t_m));
	} else if (k <= 7.8e+148) {
		tmp = 2.0 / ((((k * k) / l) * ((Math.sin(k) * t_m) / l)) * Math.tan(k));
	} else {
		tmp = (2.0 / ((t_m * Math.sin(k)) / l)) / (((((k / t_m) * (k / t_m)) * Math.tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.35e-18:
		tmp = 2.0 / (((t_m / l) * k) * ((((math.pow((k / t_m), 2.0) + 2.0) * math.tan(k)) * (t_m / l)) * t_m))
	elif k <= 7.8e+148:
		tmp = 2.0 / ((((k * k) / l) * ((math.sin(k) * t_m) / l)) * math.tan(k))
	else:
		tmp = (2.0 / ((t_m * math.sin(k)) / l)) / (((((k / t_m) * (k / t_m)) * math.tan(k)) * (t_m / l)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.35e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(k)) * Float64(t_m / l)) * t_m)));
	elseif (k <= 7.8e+148)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(Float64(sin(k) * t_m) / l)) * tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * sin(k)) / l)) / Float64(Float64(Float64(Float64(Float64(k / t_m) * Float64(k / t_m)) * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.35e-18)
		tmp = 2.0 / (((t_m / l) * k) * ((((((k / t_m) ^ 2.0) + 2.0) * tan(k)) * (t_m / l)) * t_m));
	elseif (k <= 7.8e+148)
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	else
		tmp = (2.0 / ((t_m * sin(k)) / l)) / (((((k / t_m) * (k / t_m)) * tan(k)) * (t_m / l)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.35e-18], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.8e+148], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.34999999999999994e-18

   1. Initial program 64.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 78.2

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 78.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-/.f64 74.2

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 74.2%

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t}{\ell}} \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot t\right)} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}} \]

   9. Applied rewrites 82.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}} \]

   if 1.34999999999999994e-18 < k < 7.80000000000000004e148

   1. Initial program 57.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 69.0

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 69.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-/.f64 56.0

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 56.0%

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

   9. Applied rewrites 56.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \tan k}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}} \cdot \tan k} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k} \]

       2. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{k}^{2}}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       5. unpow2 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k} \]

       10. lower-sin.f64 80.2

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k} \cdot t}{\ell}\right) \cdot \tan k} \]

   12. Applied rewrites 80.2%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \tan k} \]

   if 7.80000000000000004e148 < k

   1. Initial program 25.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 50.0

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 50.0%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 58.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. lower-/.f64 58.8

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell}}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\sin k \cdot t}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      8. lower-*.f64 58.8

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   8. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{\frac{{k}^{2}}{{t}^{2}}} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\frac{\color{blue}{k \cdot k}}{{t}^{2}} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

       2. unpow2 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\frac{k \cdot k}{\color{blue}{t \cdot t}} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

       3. times-frac N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left(\color{blue}{\frac{k}{t}} \cdot \frac{k}{t}\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

       6. lower-/.f64 64.4

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left(\frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]

   11. Applied rewrites 64.4%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \sin k}{\ell}}}{\left(\left(2 \cdot \tan k\right) \cdot \frac{t\_m}{\ell}\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.85e+20)
    (/ 2.0 (* (* (/ (* k k) l) (/ (* (sin k) t_m) l)) (tan k)))
    (/ (/ 2.0 (/ (* t_m (sin k)) l)) (* (* (* 2.0 (tan k)) (/ t_m l)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e+20) {
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	} else {
		tmp = (2.0 / ((t_m * sin(k)) / l)) / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.85d+20) then
        tmp = 2.0d0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k))
    else
        tmp = (2.0d0 / ((t_m * sin(k)) / l)) / (((2.0d0 * tan(k)) * (t_m / l)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.85e+20) {
		tmp = 2.0 / ((((k * k) / l) * ((Math.sin(k) * t_m) / l)) * Math.tan(k));
	} else {
		tmp = (2.0 / ((t_m * Math.sin(k)) / l)) / (((2.0 * Math.tan(k)) * (t_m / l)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.85e+20:
		tmp = 2.0 / ((((k * k) / l) * ((math.sin(k) * t_m) / l)) * math.tan(k))
	else:
		tmp = (2.0 / ((t_m * math.sin(k)) / l)) / (((2.0 * math.tan(k)) * (t_m / l)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.85e+20)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(Float64(sin(k) * t_m) / l)) * tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * sin(k)) / l)) / Float64(Float64(Float64(2.0 * tan(k)) * Float64(t_m / l)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.85e+20)
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	else
		tmp = (2.0 / ((t_m * sin(k)) / l)) / (((2.0 * tan(k)) * (t_m / l)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.85e+20], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e20

   1. Initial program 52.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 65.9

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-/.f64 60.5

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 60.5%

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

   9. Applied rewrites 66.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \tan k}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}} \cdot \tan k} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k} \]

       2. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{k}^{2}}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       5. unpow2 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k} \]

       10. lower-sin.f64 67.3

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k} \cdot t}{\ell}\right) \cdot \tan k} \]

   12. Applied rewrites 67.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \tan k} \]

   if 1.85e20 < t

   1. Initial program 76.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 93.1

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. lower-/.f64 98.2

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell}}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\sin k \cdot t}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      8. lower-*.f64 98.2

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{2} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.4%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\color{blue}{2} \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t\_m}{\ell}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \sin k}{\ell}}}{\left(\frac{k \cdot t\_m}{\ell} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7e+18)
    (/ 2.0 (* (* (/ (* k k) l) (/ (* (sin k) t_m) l)) (tan k)))
    (/ (/ 2.0 (/ (* t_m (sin k)) l)) (* (* (/ (* k t_m) l) 2.0) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e+18) {
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	} else {
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 7d+18) then
        tmp = 2.0d0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k))
    else
        tmp = (2.0d0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0d0) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7e+18) {
		tmp = 2.0 / ((((k * k) / l) * ((Math.sin(k) * t_m) / l)) * Math.tan(k));
	} else {
		tmp = (2.0 / ((t_m * Math.sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 7e+18:
		tmp = 2.0 / ((((k * k) / l) * ((math.sin(k) * t_m) / l)) * math.tan(k))
	else:
		tmp = (2.0 / ((t_m * math.sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7e+18)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) / l) * Float64(Float64(sin(k) * t_m) / l)) * tan(k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * sin(k)) / l)) / Float64(Float64(Float64(Float64(k * t_m) / l) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 7e+18)
		tmp = 2.0 / ((((k * k) / l) * ((sin(k) * t_m) / l)) * tan(k));
	else
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7e+18], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7e18

   1. Initial program 52.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 65.9

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 65.9%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-/.f64 60.5

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 60.5%

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

   9. Applied rewrites 66.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \tan k}} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}} \cdot \tan k} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\color{blue}{\ell \cdot \ell}} \cdot \tan k} \]

       2. times-frac N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{{k}^{2}}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       5. unpow2 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k \cdot t}}{\ell}\right) \cdot \tan k} \]

       10. lower-sin.f64 67.3

           \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\color{blue}{\sin k} \cdot t}{\ell}\right) \cdot \tan k} \]

   12. Applied rewrites 67.3%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \tan k} \]

   if 7e18 < t

   1. Initial program 76.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 93.1

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 93.1%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 98.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. lower-/.f64 98.2

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell}}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\sin k \cdot t}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      8. lower-*.f64 98.2

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(2 \cdot \frac{k \cdot t}{\ell}\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot 2\right) \cdot t} \]

       4. lower-*.f64 83.5

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\frac{\color{blue}{k \cdot t}}{\ell} \cdot 2\right) \cdot t} \]

   11. Applied rewrites 83.5%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 75.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \sin k}{\ell}}}{\left(\frac{k \cdot t\_m}{\ell} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+154)
    (/
     2.0
     (*
      (/ (* (sin k) t_m) l)
      (*
       (/
        (fma
         (* (fma 0.6666666666666666 (* t_m t_m) 1.0) k)
         k
         (* (* t_m t_m) 2.0))
        l)
       k)))
    (/ (/ 2.0 (/ (* t_m (sin k)) l)) (* (* (/ (* k t_m) l) 2.0) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+154) {
		tmp = 2.0 / (((sin(k) * t_m) / l) * ((fma((fma(0.6666666666666666, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * k));
	} else {
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * t_m) / l) * Float64(Float64(fma(Float64(fma(0.6666666666666666, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * sin(k)) / l)) / Float64(Float64(Float64(Float64(k * t_m) / l) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+154], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.6666666666666666 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000004e154

   1. Initial program 67.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 82.8

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 82.8%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 86.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(k \cdot \left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot k\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{\ell} + \frac{{k}^{2} \cdot \left({t}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}{\ell}\right) \cdot k\right)}} \]

   9. Applied rewrites 86.3%

      \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot k\right)}} \]

   if 5.00000000000000004e154 < (*.f64 l l)

   1. Initial program 43.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 62.6

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 62.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 68.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. lower-/.f64 68.4

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell}}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\sin k \cdot t}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      8. lower-*.f64 68.4

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   8. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(2 \cdot \frac{k \cdot t}{\ell}\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot 2\right) \cdot t} \]

       4. lower-*.f64 60.1

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\frac{\color{blue}{k \cdot t}}{\ell} \cdot 2\right) \cdot t} \]

   11. Applied rewrites 60.1%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 74.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m \cdot \sin k}{\ell}}}{\left(\frac{k \cdot t\_m}{\ell} \cdot 2\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.25e-68)
    (* (/ 2.0 (* (* k k) t_m)) (* l (/ l (* k k))))
    (/ (/ 2.0 (/ (* t_m (sin k)) l)) (* (* (/ (* k t_m) l) 2.0) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-68) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.25d-68) then
        tmp = (2.0d0 / ((k * k) * t_m)) * (l * (l / (k * k)))
    else
        tmp = (2.0d0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0d0) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.25e-68) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = (2.0 / ((t_m * Math.sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.25e-68:
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)))
	else:
		tmp = (2.0 / ((t_m * math.sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.25e-68)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(l * Float64(l / Float64(k * k))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m * sin(k)) / l)) / Float64(Float64(Float64(Float64(k * t_m) / l) * 2.0) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.25e-68)
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	else
		tmp = (2.0 / ((t_m * sin(k)) / l)) / ((((k * t_m) / l) * 2.0) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.25e-68], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.24999999999999993e-68

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{\sin k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}{{\sin k}^{2}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}{{\sin k}^{2}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{{\sin k}^{2}}} \]

      17. lower-sin.f64 63.4

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]

   5. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

   if 1.24999999999999993e-68 < t

   1. Initial program 69.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. cube-mult N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(t \cdot \frac{t \cdot t}{\ell \cdot \ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \color{blue}{\left(\frac{t \cdot t}{\ell \cdot \ell} \cdot \sin k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(t \cdot \left(\frac{t \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. times-frac N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-/.f64 83.5

          \[\leadsto \frac{2}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 83.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot t\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   6. Applied rewrites 93.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{\sin k \cdot t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\sin k \cdot t}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      5. lower-/.f64 93.0

         \[\leadsto \frac{\color{blue}{\frac{2}{\frac{\sin k \cdot t}{\ell}}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\sin k \cdot t}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      8. lower-*.f64 93.0

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot \sin k}}{\ell}}}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot t\right)}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   8. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right) \cdot \frac{t}{\ell}\right) \cdot t}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(2 \cdot \frac{k \cdot t}{\ell}\right)} \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\color{blue}{\frac{k \cdot t}{\ell}} \cdot 2\right) \cdot t} \]

       4. lower-*.f64 76.0

          \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\left(\frac{\color{blue}{k \cdot t}}{\ell} \cdot 2\right) \cdot t} \]

   11. Applied rewrites 76.0%

       \[\leadsto \frac{\frac{2}{\frac{t \cdot \sin k}{\ell}}}{\color{blue}{\left(\frac{k \cdot t}{\ell} \cdot 2\right)} \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 70.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right)\right) \cdot \tan k}\\ \end{array} \end{array} \]

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 2.35e-80)
    (* (/ 2.0 (* (* k k) t_m)) (* l (/ l (* k k))))
    (/ 2.0 (* (* 2.0 (* (* (/ t_m l) k) (* (/ t_m l) t_m))) (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 (t_m <= 2.35e-80) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = 2.0 / ((2.0 * (((t_m / l) * k) * ((t_m / l) * t_m))) * 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 (t_m <= 2.35d-80) then
        tmp = (2.0d0 / ((k * k) * t_m)) * (l * (l / (k * k)))
    else
        tmp = 2.0d0 / ((2.0d0 * (((t_m / l) * k) * ((t_m / l) * t_m))) * 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 (t_m <= 2.35e-80) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = 2.0 / ((2.0 * (((t_m / l) * k) * ((t_m / l) * t_m))) * 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 t_m <= 2.35e-80:
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)))
	else:
		tmp = 2.0 / ((2.0 * (((t_m / l) * k) * ((t_m / l) * t_m))) * 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 (t_m <= 2.35e-80)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(l * Float64(l / Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(Float64(Float64(t_m / l) * k) * Float64(Float64(t_m / l) * t_m))) * 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 (t_m <= 2.35e-80)
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	else
		tmp = 2.0 / ((2.0 * (((t_m / l) * k) * ((t_m / l) * t_m))) * 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[t$95$m, 2.35e-80], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.34999999999999986e-80

   1. Initial program 52.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{\sin k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}{{\sin k}^{2}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}{{\sin k}^{2}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{{\sin k}^{2}}} \]

      17. lower-sin.f64 62.8

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.2%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

   if 2.34999999999999986e-80 < t

   1. Initial program 70.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{{t}^{3}} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. unpow3 N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \sin k}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot \left(t \cdot \sin k\right)}}{\ell \cdot \ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. times-frac N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      13. lower-*.f64 86.5

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot \sin k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 86.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t \cdot \sin k}{\ell}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{\color{blue}{t \cdot k}}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lower-/.f64 76.1

         \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   7. Applied rewrites 76.1%

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot k\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right) \cdot \tan k\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot k\right)\right)\right) \cdot \tan k}} \]

   9. Applied rewrites 77.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \tan k}} \]

   10. Taylor expanded in t around inf

       \[\leadsto \frac{2}{\left(\color{blue}{2} \cdot \left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \tan k} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.2%

       \[\leadsto \frac{2}{\left(\color{blue}{2} \cdot \left(\left(\frac{t}{\ell} \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot t\right)\right)\right) \cdot \tan k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 72.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-41)
    (* (/ 2.0 (* (* k k) t_m)) (* l (/ l (* k k))))
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-41) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.5d-41) then
        tmp = (2.0d0 / ((k * k) * t_m)) * (l * (l / (k * k)))
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-41) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.5e-41:
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)))
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-41)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(l * Float64(l / Float64(k * k))));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.5e-41)
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-41], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.5e-41

   1. Initial program 51.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{\sin k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}{{\sin k}^{2}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}{{\sin k}^{2}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{{\sin k}^{2}}} \]

      17. lower-sin.f64 62.8

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

   if 4.5e-41 < t

   1. Initial program 74.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 67.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.7%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.4%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 67.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.1e-39)
    (* (/ 2.0 (* (* k k) t_m)) (* l (/ l (* k k))))
    (/ (* (/ l k) l) (* (* k (* t_m t_m)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-39) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.1d-39) then
        tmp = (2.0d0 / ((k * k) * t_m)) * (l * (l / (k * k)))
    else
        tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-39) {
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	} else {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.1e-39:
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)))
	else:
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.1e-39)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(l * Float64(l / Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m * t_m)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.1e-39)
		tmp = (2.0 / ((k * k) * t_m)) * (l * (l / (k * k)));
	else
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-39], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.0999999999999997e-39

   1. Initial program 51.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{\sin k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}{{\sin k}^{2}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}{{\sin k}^{2}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{{\sin k}^{2}}} \]

      17. lower-sin.f64 62.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.1%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right) \]

   if 3.0999999999999997e-39 < t

   1. Initial program 75.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 68.8

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 68.8%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 74.6%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{{t}^{3} \cdot k}} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.5%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 64.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.95e-147)
    (/ (* (/ l k) l) (* (* k (* t_m t_m)) t_m))
    (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.95e-147) {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.95d-147) then
        tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
    else
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.95e-147) {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.95e-147:
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
	else:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.95e-147)
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m * t_m)) * t_m));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.95e-147)
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	else
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.95e-147], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.9499999999999999e-147

   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. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 60.9

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.9%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.1%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{{t}^{3} \cdot k}} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.8%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

   if 1.9499999999999999e-147 < k

   1. Initial program 50.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 50.2

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.2%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.0%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.0%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (/ l k) l) (* (* k (* t_m t_m)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / k) * l) / ((k * (t_m * t_m)) * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / k) * l) / ((k * (t_m * t_m)) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / k) * l) / ((k * (t_m * t_m)) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l / k) * l) / ((k * (t_m * t_m)) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m * t_m)) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / k) * l) / ((k * (t_m * t_m)) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 56.5

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 56.5%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 60.6%

   \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\color{blue}{{t}^{3} \cdot k}} \]
10. Step-by-step derivation

11. Applied rewrites 64.0%

    \[\leadsto \frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
12. Add Preprocessing

Alternative 21: 58.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m \cdot t\_m} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l (/ l (* (* k k) t_m))) (* t_m t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * Float64(l / Float64(Float64(k * k) * t_m))) / Float64(t_m * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * (l / ((k * k) * t_m))) / (t_m * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 56.5

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 56.5%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 58.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 58.8%

    \[\leadsto \frac{\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}}{\color{blue}{t \cdot t}} \]
12. Add Preprocessing

Alternative 22: 58.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{\left(k \cdot k\right) \cdot t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ (/ l (* t_m t_m)) (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(Float64(l / Float64(t_m * t_m)) / Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(l * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 56.5

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 56.5%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 58.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 58.7%

    \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{t \cdot t}}{\left(k \cdot k\right) \cdot t}} \]
12. Add Preprocessing

Alternative 23: 54.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l l) (* (* t_m t_m) (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(t_m * t_m) * Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 56.5

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 56.5%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 56.5%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 58.2%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
10. Step-by-step derivation

11. Applied rewrites 55.6%

    \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))