Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 95.5%

Time: 12.5s

Alternatives: 22

Speedup: 10.3×

• 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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.62 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \left(\frac{t\_m \cdot k}{\ell} \cdot \frac{k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.62e-34)
    (/ 2.0 (* (/ (pow (sin k) 2.0) l) (* (/ (* t_m k) l) (/ k (cos k)))))
    (/
     2.0
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (* (/ t_m l) (* (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 <= 1.62e-34) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / l) * (((t_m * k) / l) * (k / cos(k))));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (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 <= 1.62d-34) then
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) / l) * (((t_m * k) / l) * (k / cos(k))))
    else
        tmp = 2.0d0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (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 <= 1.62e-34) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / l) * (((t_m * k) / l) * (k / Math.cos(k))));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * Math.sin(k))) * ((t_m / l) * (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 <= 1.62e-34:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) / l) * (((t_m * k) / l) * (k / math.cos(k))))
	else:
		tmp = 2.0 / (((t_m / l) * (t_m * math.sin(k))) * ((t_m / l) * (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 <= 1.62e-34)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / l) * Float64(Float64(Float64(t_m * k) / l) * Float64(k / cos(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * 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 <= 1.62e-34)
		tmp = 2.0 / (((sin(k) ^ 2.0) / l) * (((t_m * k) / l) * (k / cos(k))));
	else
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (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, 1.62e-34], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$m * k), $MachinePrecision] / l), $MachinePrecision] * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $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 < 1.62000000000000006e-34

   1. Initial program 54.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 65.8

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

   5. Applied rewrites 65.8%

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

   7. Applied rewrites 79.6%

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

   if 1.62000000000000006e-34 < t

   1. Initial program 66.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 95.7%

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

Alternative 2: 81.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 1720000:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k \cdot \ell} \cdot \frac{\left(t\_m \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-107)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 1720000.0)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (/ 2.0 (* (/ k (* (cos k) l)) (/ (* (* t_m k) (pow (sin k) 2.0)) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-107) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 1720000.0) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / ((k / (cos(k) * l)) * (((t_m * k) * pow(sin(k), 2.0)) / l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-107)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 1720000.0)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / Float64(cos(k) * l)) * Float64(Float64(Float64(t_m * k) * (sin(k) ^ 2.0)) / l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6e-107], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1720000.0], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * k), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.9999999999999994e-107

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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 63.2%

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

   9. Applied rewrites 78.1%

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

   if 5.9999999999999994e-107 < k < 1.72e6

   1. Initial program 48.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.9

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 87.0%

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

   if 1.72e6 < k

   1. Initial program 44.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 90.7%

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

Alternative 3: 80.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 1720000:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot k\right) \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-107)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 1720000.0)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (/ 2.0 (/ (* (* (* (/ (sin k) l) (tan k)) k) (* t_m k)) l))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-107) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 1720000.0) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / (((((sin(k) / l) * tan(k)) * k) * (t_m * k)) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-107)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 1720000.0)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * k) * Float64(t_m * k)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6e-107], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1720000.0], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.9999999999999994e-107

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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 63.2%

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

   9. Applied rewrites 78.1%

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

   if 5.9999999999999994e-107 < k < 1.72e6

   1. Initial program 48.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.9

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 87.0%

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

   if 1.72e6 < k

   1. Initial program 44.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 90.6%

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

Alternative 4: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 1720000:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sin k}{\ell} \cdot \left(\tan k \cdot \frac{\left(t\_m \cdot k\right) \cdot k}{\ell}\right)}\\ \end{array} \end{array} \]

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 6e-107)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 1720000.0)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (/ 2.0 (* (/ (sin k) l) (* (tan k) (/ (* (* t_m k) k) l))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-107) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 1720000.0) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / ((sin(k) / l) * (tan(k) * (((t_m * k) * k) / l)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-107)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 1720000.0)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) / l) * Float64(tan(k) * Float64(Float64(Float64(t_m * k) * k) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6e-107], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1720000.0], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.9999999999999994e-107

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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 63.2%

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

   9. Applied rewrites 78.1%

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

   if 5.9999999999999994e-107 < k < 1.72e6

   1. Initial program 48.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.9

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 87.0%

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

   if 1.72e6 < k

   1. Initial program 44.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 86.8%

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

Alternative 5: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 10000000:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \tan k\right) \cdot \left(\left(t\_m \cdot k\right) \cdot k\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-107)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 10000000.0)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (/ 2.0 (/ (* (* (sin k) (tan k)) (* (* t_m k) k)) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-107) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 10000000.0) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / (((sin(k) * tan(k)) * ((t_m * k) * k)) / (l * l));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-107)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 10000000.0)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t_m * k) * k)) / Float64(l * l)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6e-107], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10000000.0], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.9999999999999994e-107

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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 63.2%

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

   9. Applied rewrites 78.1%

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

   if 5.9999999999999994e-107 < k < 1e7

   1. Initial program 48.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.9

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 87.0%

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

   if 1e7 < k

   1. Initial program 44.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 82.8%

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

Alternative 6: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 10000000:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\frac{t\_m}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6e-107)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 10000000.0)
      (/
       (/ 2.0 (/ t_m l))
       (*
        (/
         (fma
          (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
          k
          (* (* 2.0 t_m) t_m))
         l)
        (* k k)))
      (/ 2.0 (* k (* k (* (/ t_m (* l l)) (* (tan k) (sin k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6e-107) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 10000000.0) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = 2.0 / (k * (k * ((t_m / (l * l)) * (tan(k) * sin(k)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6e-107)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 10000000.0)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(t_m / Float64(l * l)) * Float64(tan(k) * sin(k))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6e-107], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10000000.0], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(t$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.9999999999999994e-107

   1. Initial program 62.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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 63.2%

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

   9. Applied rewrites 78.1%

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

   if 5.9999999999999994e-107 < k < 1e7

   1. Initial program 48.6%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 83.9

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 87.0%

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

   if 1e7 < k

   1. Initial program 44.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 77.0%

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

Alternative 7: 84.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\tan k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot k\right) \cdot \left(t\_m \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 8e-30)
    (/ 2.0 (* (* (* (/ (tan k) l) (/ (sin k) l)) k) (* t_m k)))
    (* (/ (/ l t_m) (* k t_m)) (/ l (* 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 (t_m <= 8e-30) {
		tmp = 2.0 / ((((tan(k) / l) * (sin(k) / l)) * k) * (t_m * k));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 8d-30) then
        tmp = 2.0d0 / ((((tan(k) / l) * (sin(k) / l)) * k) * (t_m * k))
    else
        tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 8e-30) {
		tmp = 2.0 / ((((Math.tan(k) / l) * (Math.sin(k) / l)) * k) * (t_m * k));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 t_m <= 8e-30:
		tmp = 2.0 / ((((math.tan(k) / l) * (math.sin(k) / l)) * k) * (t_m * k))
	else:
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 8e-30)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(tan(k) / l) * Float64(sin(k) / l)) * k) * Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e-30)
		tmp = 2.0 / ((((tan(k) / l) * (sin(k) / l)) * k) * (t_m * k));
	else
		tmp = ((l / t_m) / (k * t_m)) * (l / (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[t$95$m, 8e-30], N[(2.0 / N[(N[(N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.000000000000001e-30

   1. Initial program 54.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 72.8%

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

   if 8.000000000000001e-30 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 67.8%

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

   9. Applied rewrites 87.5%

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

Alternative 8: 84.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t\_m \cdot k\right) \cdot \left(\frac{\tan k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 8e-30)
    (/ 2.0 (* k (* (* t_m k) (* (/ (tan k) l) (/ (sin k) l)))))
    (* (/ (/ l t_m) (* k t_m)) (/ l (* 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 (t_m <= 8e-30) {
		tmp = 2.0 / (k * ((t_m * k) * ((tan(k) / l) * (sin(k) / l))));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 8d-30) then
        tmp = 2.0d0 / (k * ((t_m * k) * ((tan(k) / l) * (sin(k) / l))))
    else
        tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 8e-30) {
		tmp = 2.0 / (k * ((t_m * k) * ((Math.tan(k) / l) * (Math.sin(k) / l))));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 t_m <= 8e-30:
		tmp = 2.0 / (k * ((t_m * k) * ((math.tan(k) / l) * (math.sin(k) / l))))
	else:
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 8e-30)
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t_m * k) * Float64(Float64(tan(k) / l) * Float64(sin(k) / l)))));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 8e-30)
		tmp = 2.0 / (k * ((t_m * k) * ((tan(k) / l) * (sin(k) / l))));
	else
		tmp = ((l / t_m) / (k * t_m)) * (l / (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[t$95$m, 8e-30], N[(2.0 / N[(k * N[(N[(t$95$m * k), $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.000000000000001e-30

   1. Initial program 54.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 65.9

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 72.9%

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

   if 8.000000000000001e-30 < t

   1. Initial program 66.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 63.4

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

   5. Applied rewrites 63.4%

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

   7. Applied rewrites 67.8%

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

   9. Applied rewrites 87.5%

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

Alternative 9: 77.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(2 \cdot t\_m\right) \cdot t\_m\right)}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 1.2e-43)
    (/
     (/ 2.0 (/ t_m l))
     (*
      (/
       (fma
        (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
        k
        (* (* 2.0 t_m) t_m))
       l)
      (* k k)))
    (* (/ (/ l t_m) (* k t_m)) (/ l (* 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 (t_m <= 1.2e-43) {
		tmp = (2.0 / (t_m / l)) / ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((2.0 * t_m) * t_m)) / l) * (k * k));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 1.2e-43)
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(2.0 * t_m) * t_m)) / l) * Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-43], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(2.0 * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2000000000000001e-43

   1. Initial program 53.9%

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

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 79.7

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

   8. Applied rewrites 79.7%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 69.6%

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

   if 1.2000000000000001e-43 < t

   1. Initial program 67.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.9

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

   5. Applied rewrites 64.9%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 88.0%

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

Alternative 10: 72.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{-\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m} \cdot \frac{\ell}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6500000000000.0)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 4.4e+129)
      (* (/ l t_m) (/ (- l) (* (* t_m (* t_m k)) k)))
      (/ (* (/ l (* (* k k) t_m)) (/ l t_m)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 4.4e+129) {
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	} else {
		tmp = ((l / ((k * k) * t_m)) * (l / t_m)) / t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6500000000000.0d0) then
        tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m))
    else if (k <= 4.4d+129) then
        tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k))
    else
        tmp = ((l / ((k * k) * t_m)) * (l / t_m)) / t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 4.4e+129) {
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	} else {
		tmp = ((l / ((k * k) * t_m)) * (l / t_m)) / t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6500000000000.0:
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m))
	elif k <= 4.4e+129:
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k))
	else:
		tmp = ((l / ((k * k) * t_m)) * (l / t_m)) / t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6500000000000.0)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 4.4e+129)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(-l) / Float64(Float64(t_m * Float64(t_m * k)) * k)));
	else
		tmp = Float64(Float64(Float64(l / Float64(Float64(k * k) * t_m)) * Float64(l / t_m)) / t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6500000000000.0)
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	elseif (k <= 4.4e+129)
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	else
		tmp = ((l / ((k * k) * t_m)) * (l / t_m)) / t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6500000000000.0], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.4e+129], N[(N[(l / t$95$m), $MachinePrecision] * N[((-l) / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.5e12

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 77.2%

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

   if 6.5e12 < k < 4.3999999999999999e129

   1. Initial program 61.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 45.7%

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

   9. Applied rewrites 58.2%

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

   11. Applied rewrites 58.2%

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

   if 4.3999999999999999e129 < k

   1. Initial program 33.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 30.8

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 34.7%

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

   9. Applied rewrites 63.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{-\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.25 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

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-45)
    (/ 2.0 (* (* (* k k) t_m) (* (/ k l) (/ k l))))
    (* (/ (/ l t_m) (* k t_m)) (/ l (* 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 (t_m <= 1.25e-45) {
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 1.25d-45) then
        tmp = 2.0d0 / (((k * k) * t_m) * ((k / l) * (k / l)))
    else
        tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 1.25e-45) {
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)));
	} else {
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 t_m <= 1.25e-45:
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)))
	else:
		tmp = ((l / t_m) / (k * t_m)) * (l / (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 (t_m <= 1.25e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(k / l) * Float64(k / l))));
	else
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.25e-45)
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)));
	else
		tmp = ((l / t_m) / (k * t_m)) * (l / (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[t$95$m, 1.25e-45], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.24999999999999994e-45

   1. Initial program 53.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 65.4

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

   5. Applied rewrites 65.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 58.4%

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

   if 1.24999999999999994e-45 < t

   1. Initial program 67.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.9

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

   5. Applied rewrites 64.9%

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

   7. Applied rewrites 69.1%

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

   9. Applied rewrites 88.0%

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

Alternative 12: 72.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\frac{\ell}{t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{-\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6500000000000.0)
    (* (/ (/ l t_m) (* k t_m)) (/ l (* k t_m)))
    (if (<= k 4e+130)
      (* (/ l t_m) (/ (- l) (* (* t_m (* t_m k)) k)))
      (* (/ l t_m) (/ l (* (* (* k k) t_m) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 4e+130) {
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6500000000000.0d0) then
        tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m))
    else if (k <= 4d+130) then
        tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k))
    else
        tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	} else if (k <= 4e+130) {
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6500000000000.0:
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m))
	elif k <= 4e+130:
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k))
	else:
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6500000000000.0)
		tmp = Float64(Float64(Float64(l / t_m) / Float64(k * t_m)) * Float64(l / Float64(k * t_m)));
	elseif (k <= 4e+130)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(-l) / Float64(Float64(t_m * Float64(t_m * k)) * k)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6500000000000.0)
		tmp = ((l / t_m) / (k * t_m)) * (l / (k * t_m));
	elseif (k <= 4e+130)
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	else
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6500000000000.0], N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+130], N[(N[(l / t$95$m), $MachinePrecision] * N[((-l) / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.5e12

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 77.2%

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

   if 6.5e12 < k < 4.0000000000000002e130

   1. Initial program 61.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 45.7%

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

   9. Applied rewrites 58.2%

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

   11. Applied rewrites 58.2%

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

   if 4.0000000000000002e130 < k

   1. Initial program 33.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 30.8

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 49.7%

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

   9. Applied rewrites 59.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{-\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{k \cdot t\_m}}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{-\ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6500000000000.0)
    (* (/ l t_m) (/ (/ l (* k t_m)) (* k t_m)))
    (if (<= k 4e+130)
      (* (/ l t_m) (/ (- l) (* (* t_m (* t_m k)) k)))
      (* (/ l t_m) (/ l (* (* (* k k) t_m) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = (l / t_m) * ((l / (k * t_m)) / (k * t_m));
	} else if (k <= 4e+130) {
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6500000000000.0d0) then
        tmp = (l / t_m) * ((l / (k * t_m)) / (k * t_m))
    else if (k <= 4d+130) then
        tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k))
    else
        tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = (l / t_m) * ((l / (k * t_m)) / (k * t_m));
	} else if (k <= 4e+130) {
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6500000000000.0:
		tmp = (l / t_m) * ((l / (k * t_m)) / (k * t_m))
	elif k <= 4e+130:
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k))
	else:
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6500000000000.0)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(k * t_m)) / Float64(k * t_m)));
	elseif (k <= 4e+130)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(-l) / Float64(Float64(t_m * Float64(t_m * k)) * k)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6500000000000.0)
		tmp = (l / t_m) * ((l / (k * t_m)) / (k * t_m));
	elseif (k <= 4e+130)
		tmp = (l / t_m) * (-l / ((t_m * (t_m * k)) * k));
	else
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6500000000000.0], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+130], N[(N[(l / t$95$m), $MachinePrecision] * N[((-l) / N[(N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.5e12

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 75.9%

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

   9. Applied rewrites 76.4%

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

   if 6.5e12 < k < 4.0000000000000002e130

   1. Initial program 61.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 45.7%

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

   9. Applied rewrites 58.2%

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

   11. Applied rewrites 58.2%

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

   if 4.0000000000000002e130 < k

   1. Initial program 33.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 30.8

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 49.7%

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

   9. Applied rewrites 59.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\ell}{k \cdot t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{-\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(t\_m \cdot k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\ell}{t\_2} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{-\ell}{t\_2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* t_m k))))
   (*
    t_s
    (if (<= k 6500000000000.0)
      (* (/ l t_2) (/ l (* k t_m)))
      (if (<= k 4e+130)
        (* (/ l t_m) (/ (- l) (* t_2 k)))
        (* (/ l t_m) (/ l (* (* (* k k) t_m) t_m))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (t_m * k);
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = (l / t_2) * (l / (k * t_m));
	} else if (k <= 4e+130) {
		tmp = (l / t_m) * (-l / (t_2 * k));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (t_m * k)
    if (k <= 6500000000000.0d0) then
        tmp = (l / t_2) * (l / (k * t_m))
    else if (k <= 4d+130) then
        tmp = (l / t_m) * (-l / (t_2 * k))
    else
        tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * (t_m * k);
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = (l / t_2) * (l / (k * t_m));
	} else if (k <= 4e+130) {
		tmp = (l / t_m) * (-l / (t_2 * k));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * (t_m * k)
	tmp = 0
	if k <= 6500000000000.0:
		tmp = (l / t_2) * (l / (k * t_m))
	elif k <= 4e+130:
		tmp = (l / t_m) * (-l / (t_2 * k))
	else:
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m * Float64(t_m * k))
	tmp = 0.0
	if (k <= 6500000000000.0)
		tmp = Float64(Float64(l / t_2) * Float64(l / Float64(k * t_m)));
	elseif (k <= 4e+130)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(-l) / Float64(t_2 * k)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * (t_m * k);
	tmp = 0.0;
	if (k <= 6500000000000.0)
		tmp = (l / t_2) * (l / (k * t_m));
	elseif (k <= 4e+130)
		tmp = (l / t_m) * (-l / (t_2 * k));
	else
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6500000000000.0], N[(N[(l / t$95$2), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e+130], N[(N[(l / t$95$m), $MachinePrecision] * N[((-l) / N[(t$95$2 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 6.5e12

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 69.5%

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

   11. Applied rewrites 75.8%

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

   if 6.5e12 < k < 4.0000000000000002e130

   1. Initial program 61.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 45.7%

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

   9. Applied rewrites 58.2%

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

   11. Applied rewrites 58.2%

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

   if 4.0000000000000002e130 < k

   1. Initial program 33.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 30.8

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 49.7%

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

   9. Applied rewrites 59.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\ell}{t \cdot \left(t \cdot k\right)} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{-\ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-166}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+73}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \ell}{\left(t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= k 4.3e-166)
    (/ (* l l) (* (* t_m t_m) (* (* k t_m) k)))
    (if (<= k 6e+73)
      (* l (/ (/ l (* (* k k) t_m)) (* t_m t_m)))
      (/ (* (- 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) {
	double tmp;
	if (k <= 4.3e-166) {
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k));
	} else if (k <= 6e+73) {
		tmp = l * ((l / ((k * k) * t_m)) / (t_m * t_m));
	} else {
		tmp = (-l * l) / ((t_m * ((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 <= 4.3d-166) then
        tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k))
    else if (k <= 6d+73) then
        tmp = l * ((l / ((k * k) * t_m)) / (t_m * t_m))
    else
        tmp = (-l * l) / ((t_m * ((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 <= 4.3e-166) {
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k));
	} else if (k <= 6e+73) {
		tmp = l * ((l / ((k * k) * t_m)) / (t_m * t_m));
	} else {
		tmp = (-l * l) / ((t_m * ((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 <= 4.3e-166:
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k))
	elif k <= 6e+73:
		tmp = l * ((l / ((k * k) * t_m)) / (t_m * t_m))
	else:
		tmp = (-l * l) / ((t_m * ((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 <= 4.3e-166)
		tmp = Float64(Float64(l * l) / Float64(Float64(t_m * t_m) * Float64(Float64(k * t_m) * k)));
	elseif (k <= 6e+73)
		tmp = Float64(l * Float64(Float64(l / Float64(Float64(k * k) * t_m)) / Float64(t_m * t_m)));
	else
		tmp = Float64(Float64(Float64(-l) * l) / Float64(Float64(t_m * Float64(Float64(t_m * 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 <= 4.3e-166)
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k));
	elseif (k <= 6e+73)
		tmp = l * ((l / ((k * k) * t_m)) / (t_m * t_m));
	else
		tmp = (-l * l) / ((t_m * ((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, 4.3e-166], N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e+73], N[(l * N[(N[(l / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-l) * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 4.3000000000000001e-166

   1. Initial program 62.3%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 60.9%

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

   if 4.3000000000000001e-166 < k < 6.00000000000000021e73

   1. Initial program 50.7%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   7. Applied rewrites 58.0%

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

   9. Applied rewrites 58.0%

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

   if 6.00000000000000021e73 < k

   1. Initial program 44.5%

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

   3. Taylor expanded in 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 37.1

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

   5. Applied rewrites 37.1%

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

   7. Applied rewrites 40.0%

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

   9. Applied rewrites 42.6%

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

   11. Applied rewrites 59.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-166}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+73}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\left(k \cdot k\right) \cdot t}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \ell}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-135}:\\ \;\;\;\;\frac{\ell}{t\_m \cdot \left(t\_m \cdot k\right)} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4e-135)
    (* (/ l (* t_m (* t_m k))) (/ l (* k t_m)))
    (* (/ l t_m) (/ l (* (* (* k k) t_m) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4e-135) {
		tmp = (l / (t_m * (t_m * k))) * (l / (k * t_m));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4d-135) then
        tmp = (l / (t_m * (t_m * k))) * (l / (k * t_m))
    else
        tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4e-135) {
		tmp = (l / (t_m * (t_m * k))) * (l / (k * t_m));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 4e-135:
		tmp = (l / (t_m * (t_m * k))) * (l / (k * t_m))
	else:
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4e-135)
		tmp = Float64(Float64(l / Float64(t_m * Float64(t_m * k))) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 4e-135)
		tmp = (l / (t_m * (t_m * k))) * (l / (k * t_m));
	else
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 4e-135], N[(N[(l / N[(t$95$m * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.0000000000000002e-135

   1. Initial program 62.6%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 71.3%

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

   11. Applied rewrites 76.5%

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

   if 4.0000000000000002e-135 < k

   1. Initial program 46.4%

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

   3. 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 43.1

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

   5. Applied rewrites 43.1%

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

   7. Applied rewrites 58.6%

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

   9. Applied rewrites 62.2%

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

Alternative 17: 68.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{+177}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.15e+177)
    (* (/ l t_m) (/ l (* (* k t_m) (* k t_m))))
    (* (/ l t_m) (/ l (* (* (* k k) t_m) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e+177) {
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.15d+177) then
        tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)))
    else
        tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e+177) {
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)));
	} else {
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.15e+177:
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)))
	else:
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.15e+177)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(Float64(k * k) * t_m) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.15e+177)
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)));
	else
		tmp = (l / t_m) * (l / (((k * k) * t_m) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.15e+177], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e177

   1. Initial program 59.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 72.6%

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

   9. Applied rewrites 72.6%

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

   if 1.15e177 < k

   1. Initial program 36.1%

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

   3. Taylor expanded in k around 0

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

      1. 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 36.3

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

   5. Applied rewrites 36.3%

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

   7. Applied rewrites 48.7%

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

   9. Applied rewrites 69.9%

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

Alternative 18: 68.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\ell}{\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \ell}{\left(t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= k 2.6e+48)
    (* (/ l t_m) (/ l (* (* k t_m) (* k t_m))))
    (/ (* (- 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) {
	double tmp;
	if (k <= 2.6e+48) {
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)));
	} else {
		tmp = (-l * l) / ((t_m * ((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 <= 2.6d+48) then
        tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)))
    else
        tmp = (-l * l) / ((t_m * ((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 <= 2.6e+48) {
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)));
	} else {
		tmp = (-l * l) / ((t_m * ((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 <= 2.6e+48:
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)))
	else:
		tmp = (-l * l) / ((t_m * ((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 <= 2.6e+48)
		tmp = Float64(Float64(l / t_m) * Float64(l / Float64(Float64(k * t_m) * Float64(k * t_m))));
	else
		tmp = Float64(Float64(Float64(-l) * l) / Float64(Float64(t_m * Float64(Float64(t_m * 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 <= 2.6e+48)
		tmp = (l / t_m) * (l / ((k * t_m) * (k * t_m)));
	else
		tmp = (-l * l) / ((t_m * ((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, 2.6e+48], N[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-l) * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.59999999999999995e48

   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. 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 60.0

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

   5. Applied rewrites 60.0%

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

   7. Applied rewrites 75.8%

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

   9. Applied rewrites 75.8%

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

   if 2.59999999999999995e48 < k

   1. Initial program 43.1%

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

   3. Taylor expanded in k around 0

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

      1. 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 34.6

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

   5. Applied rewrites 34.6%

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

   7. Applied rewrites 37.2%

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

   9. Applied rewrites 39.1%

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

   11. Applied rewrites 53.9%

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

Alternative 19: 58.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \ell}{\left(t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot k\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= k 6500000000000.0)
    (/ (* l l) (* (* t_m t_m) (* (* k t_m) k)))
    (/ (* (- 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) {
	double tmp;
	if (k <= 6500000000000.0) {
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k));
	} else {
		tmp = (-l * l) / ((t_m * ((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 <= 6500000000000.0d0) then
        tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k))
    else
        tmp = (-l * l) / ((t_m * ((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 <= 6500000000000.0) {
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k));
	} else {
		tmp = (-l * l) / ((t_m * ((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 <= 6500000000000.0:
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k))
	else:
		tmp = (-l * l) / ((t_m * ((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 <= 6500000000000.0)
		tmp = Float64(Float64(l * l) / Float64(Float64(t_m * t_m) * Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(Float64(Float64(-l) * l) / Float64(Float64(t_m * Float64(Float64(t_m * 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 <= 6500000000000.0)
		tmp = (l * l) / ((t_m * t_m) * ((k * t_m) * k));
	else
		tmp = (-l * l) / ((t_m * ((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, 6500000000000.0], N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-l) * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.5e12

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 59.9

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

   5. Applied rewrites 59.9%

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

   7. Applied rewrites 62.8%

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

   9. Applied rewrites 60.4%

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

   if 6.5e12 < k

   1. Initial program 44.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 36.6

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 39.1%

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

   9. Applied rewrites 40.9%

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

   11. Applied rewrites 56.9%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6500000000000:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot t\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\ell\right) \cdot \ell}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\left(-\ell\right) \cdot \ell}{\left(t\_m \cdot \left(\left(t\_m \cdot k\right) \cdot k\right)\right) \cdot t\_m} \end{array} \]

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) 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(Float64(-l) * l) / Float64(Float64(t_m * Float64(Float64(t_m * 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 * N[(N[(t$95$m * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

3. 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 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 58.6%

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

9. Applied rewrites 56.9%

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

11. Applied rewrites 35.2%

    \[\leadsto \frac{\left(-\ell\right) \cdot \ell}{\left(t \cdot \left(\left(t \cdot k\right) \cdot k\right)\right) \cdot \color{blue}{t}} \]
12. Add Preprocessing

Alternative 21: 35.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\left(-\ell\right) \cdot \ell}{\left(t\_m \cdot \left(t\_m \cdot k\right)\right) \cdot \left(t\_m \cdot k\right)} \end{array} \]

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)) (* t_m k)))))

C

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

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((-l * l) / ((t_m * (t_m * k)) * (t_m * k)))
end function

Java

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

Python

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

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(-l) * l) / Float64(Float64(t_m * Float64(t_m * k)) * Float64(t_m * k))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.8%

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

3. 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 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 58.6%

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

9. Applied rewrites 56.9%

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

11. Applied rewrites 34.5%

    \[\leadsto \frac{\left(-\ell\right) \cdot \ell}{\left(t \cdot \left(t \cdot k\right)\right) \cdot \color{blue}{\left(t \cdot k\right)}} \]
12. Add Preprocessing

Alternative 22: 31.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\left(-\ell\right) \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

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(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 57.8%

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

3. Taylor expanded in k around 0

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

   1. unpow2 N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 58.6%

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

9. Applied rewrites 56.9%

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

11. Applied rewrites 28.7%

    \[\leadsto \frac{\left(-\ell\right) \cdot \ell}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{t}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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))))