Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 90.5%

Time: 18.2s

Alternatives: 16

Speedup: 32.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999998e-82

   1. Initial program 52.0%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 63.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 63.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 68.6%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\cos k}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{\sin k}^{2}}{\cos k}\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{{k}^{2} \cdot t}{\ell \cdot \ell}\right), \left(\frac{{\sin k}^{\color{blue}{2}}}{\cos k}\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\frac{{k}^{2} \cdot t}{\ell}}{\ell}\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{\cos k}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot t}{\ell}\right), \ell\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{\cos k}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \ell\right), \ell\right), \left(\frac{{\color{blue}{\sin k}}^{2}}{\cos k}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \ell\right), \ell\right), \left(\frac{{\sin \color{blue}{k}}^{2}}{\cos k}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right), \ell\right), \left(\frac{{\sin k}^{2}}{\cos k}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right), \left(\frac{{\sin k}^{2}}{\cos k}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right), \mathsf{/.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\cos k}\right)\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \cos \color{blue}{k}\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \cos k\right)\right)\right) \]

      14. cos-lowering-cos.f64 74.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{cos.f64}\left(k\right)\right)\right)\right) \]

   9. Simplified 74.9%

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

   if 4.4999999999999998e-82 < t

   1. Initial program 70.9%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 89.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 89.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 94.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)}\right), \color{blue}{2}\right) \]

   8. Applied egg-rr 94.8%

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

4. Final simplification 81.0%

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

Alternative 2: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.25 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \left(k \cdot k\right)\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \sin k}}{\frac{\left(2 + \frac{k}{\frac{t\_m}{\frac{k}{t\_m}}}\right) \cdot \tan k}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.25e-82)
    (/ 2.0 (* (* t_m (* k k)) (/ (pow (sin k) 2.0) (* l (* l (cos k))))))
    (*
     2.0
     (/
      (/ (/ l t_m) (* t_m (sin k)))
      (/ (* (+ 2.0 (/ k (/ t_m (/ k t_m)))) (tan k)) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.25e-82) {
		tmp = 2.0 / ((t_m * (k * k)) * (pow(sin(k), 2.0) / (l * (l * cos(k)))));
	} else {
		tmp = 2.0 * (((l / t_m) / (t_m * sin(k))) / (((2.0 + (k / (t_m / (k / t_m)))) * tan(k)) / (l / t_m)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.25d-82) then
        tmp = 2.0d0 / ((t_m * (k * k)) * ((sin(k) ** 2.0d0) / (l * (l * cos(k)))))
    else
        tmp = 2.0d0 * (((l / t_m) / (t_m * sin(k))) / (((2.0d0 + (k / (t_m / (k / t_m)))) * tan(k)) / (l / t_m)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.25e-82) {
		tmp = 2.0 / ((t_m * (k * k)) * (Math.pow(Math.sin(k), 2.0) / (l * (l * Math.cos(k)))));
	} else {
		tmp = 2.0 * (((l / t_m) / (t_m * Math.sin(k))) / (((2.0 + (k / (t_m / (k / t_m)))) * Math.tan(k)) / (l / t_m)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.25e-82:
		tmp = 2.0 / ((t_m * (k * k)) * (math.pow(math.sin(k), 2.0) / (l * (l * math.cos(k)))))
	else:
		tmp = 2.0 * (((l / t_m) / (t_m * math.sin(k))) / (((2.0 + (k / (t_m / (k / t_m)))) * math.tan(k)) / (l / t_m)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.25e-82)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(k * k)) * Float64((sin(k) ^ 2.0) / Float64(l * Float64(l * cos(k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t_m) / Float64(t_m * sin(k))) / Float64(Float64(Float64(2.0 + Float64(k / Float64(t_m / Float64(k / t_m)))) * tan(k)) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.25e-82)
		tmp = 2.0 / ((t_m * (k * k)) * ((sin(k) ^ 2.0) / (l * (l * cos(k)))));
	else
		tmp = 2.0 * (((l / t_m) / (t_m * sin(k))) / (((2.0 + (k / (t_m / (k / t_m)))) * tan(k)) / (l / t_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.25e-82], N[(2.0 / N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 + N[(k / N[(t$95$m / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2499999999999999e-82

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{{\ell}^{2}} \cdot \cos k}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left({k}^{2} \cdot t\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot t\right), \color{blue}{\left(\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(t \cdot {k}^{2}\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{2}\right)\right), \left(\frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(k \cdot k\right)\right), \left(\frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2} \cdot \cos k}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{{\sin k}^{\color{blue}{2}}}{{\ell}^{2} \cdot \cos k}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left({\sin k}^{2}\right), \color{blue}{\left({\ell}^{2} \cdot \cos k\right)}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\sin k, 2\right), \left(\color{blue}{{\ell}^{2}} \cdot \cos k\right)\right)\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left({\color{blue}{\ell}}^{2} \cdot \cos k\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(\left(\ell \cdot \ell\right) \cdot \cos \color{blue}{k}\right)\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \left(\ell \cdot \color{blue}{\left(\ell \cdot \cos k\right)}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(\ell \cdot \cos k\right)}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \color{blue}{\cos k}\right)\right)\right)\right)\right) \]

      15. cos-lowering-cos.f64 67.1%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right), \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{cos.f64}\left(k\right)\right)\right)\right)\right)\right) \]

   5. Simplified 67.1%

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

   if 2.2499999999999999e-82 < t

   1. Initial program 70.9%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 89.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 89.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 94.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)}\right), \color{blue}{2}\right) \]

   8. Applied egg-rr 94.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.25 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \frac{{\sin k}^{2}}{\ell \cdot \left(\ell \cdot \cos k\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \sin k}}{\frac{\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot \frac{t\_m}{\ell}\right) \cdot \frac{2 \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{t\_m}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\tan k \cdot \frac{t\_2}{\frac{\ell}{t\_m}}\right)\right)}{t\_m}}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* t_m (sin k))))
   (*
    t_s
    (if (<= k 5.4e-31)
      (/ 2.0 (* (* t_2 (/ t_m l)) (/ (* 2.0 (* t_m k)) l)))
      (if (<= k 1.35e+143)
        (/
         2.0
         (/
          (/ (* (/ t_m l) (* (* k k) (* (tan k) (/ t_2 (/ l t_m))))) t_m)
          t_m))
        (/ 2.0 (* (* k 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 t_2 = t_m * sin(k);
	double tmp;
	if (k <= 5.4e-31) {
		tmp = 2.0 / ((t_2 * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	} else if (k <= 1.35e+143) {
		tmp = 2.0 / ((((t_m / l) * ((k * k) * (tan(k) * (t_2 / (l / t_m))))) / t_m) / t_m);
	} else {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * sin(k)
    if (k <= 5.4d-31) then
        tmp = 2.0d0 / ((t_2 * (t_m / l)) * ((2.0d0 * (t_m * k)) / l))
    else if (k <= 1.35d+143) then
        tmp = 2.0d0 / ((((t_m / l) * ((k * k) * (tan(k) * (t_2 / (l / t_m))))) / t_m) / t_m)
    else
        tmp = 2.0d0 / ((k * k) * (((t_m * (k * k)) / l) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m * Math.sin(k);
	double tmp;
	if (k <= 5.4e-31) {
		tmp = 2.0 / ((t_2 * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	} else if (k <= 1.35e+143) {
		tmp = 2.0 / ((((t_m / l) * ((k * k) * (Math.tan(k) * (t_2 / (l / t_m))))) / t_m) / t_m);
	} else {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = t_m * math.sin(k)
	tmp = 0
	if k <= 5.4e-31:
		tmp = 2.0 / ((t_2 * (t_m / l)) * ((2.0 * (t_m * k)) / l))
	elif k <= 1.35e+143:
		tmp = 2.0 / ((((t_m / l) * ((k * k) * (math.tan(k) * (t_2 / (l / t_m))))) / t_m) / t_m)
	else:
		tmp = 2.0 / ((k * 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)
	t_2 = Float64(t_m * sin(k))
	tmp = 0.0
	if (k <= 5.4e-31)
		tmp = Float64(2.0 / Float64(Float64(t_2 * Float64(t_m / l)) * Float64(Float64(2.0 * Float64(t_m * k)) / l)));
	elseif (k <= 1.35e+143)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(Float64(k * k) * Float64(tan(k) * Float64(t_2 / Float64(l / t_m))))) / t_m) / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(t_m * Float64(k * k)) / l) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = t_m * sin(k);
	tmp = 0.0;
	if (k <= 5.4e-31)
		tmp = 2.0 / ((t_2 * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	elseif (k <= 1.35e+143)
		tmp = 2.0 / ((((t_m / l) * ((k * k) * (tan(k) * (t_2 / (l / t_m))))) / t_m) / t_m);
	else
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 5.4e-31], N[(2.0 / N[(N[(t$95$2 * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.35e+143], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 5.40000000000000027e-31

   1. Initial program 61.2%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 77.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 77.3%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 82.0%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(2 \cdot \frac{k \cdot t}{\ell}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2 \cdot \left(k \cdot t\right)}{\color{blue}{\ell}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(k \cdot t\right)\right), \color{blue}{\ell}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 79.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \ell\right)\right)\right) \]

   9. Simplified 79.2%

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

   if 5.40000000000000027e-31 < k < 1.3500000000000001e143

   1. Initial program 53.5%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 62.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 62.8%

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

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2}}{{t}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left({k}^{2}\right), \color{blue}{\left({t}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(k \cdot k\right), \left({\color{blue}{t}}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({\color{blue}{t}}^{2}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(t \cdot \color{blue}{t}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 28.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, \color{blue}{t}\right)\right)\right)\right) \]

   7. Simplified 28.7%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(k \cdot k\right)}{\color{blue}{t \cdot t}}\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{\frac{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(k \cdot k\right)}{t}}{\color{blue}{t}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \tan k\right) \cdot \left(k \cdot k\right)}{t}\right), \color{blue}{t}\right)\right) \]

   9. Applied egg-rr 87.6%

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

   if 1.3500000000000001e143 < k

   1. Initial program 48.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 55.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{{k}^{2} \cdot t}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\frac{{k}^{2} \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \ell\right), \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \ell\right), \ell\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right), \ell\right)\right)\right) \]

      7. *-lowering-*.f64 69.4%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right)\right)\right) \]

   8. Simplified 69.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot \left(t \cdot k\right)}{\ell}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{\frac{\frac{\frac{t}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \left(\tan k \cdot \frac{t \cdot \sin k}{\frac{\ell}{t}}\right)\right)}{t}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot \left(t\_m \cdot t\_m\right)}{\ell}, \frac{2}{\ell}, \frac{k}{\ell} \cdot \frac{k \cdot \left(t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333 + 1\right)\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot \sin k}}{\frac{\left(2 + \frac{k}{\frac{t\_m}{\frac{k}{t\_m}}}\right) \cdot \tan k}{\frac{\ell}{t\_m}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-84)
    (/
     2.0
     (*
      (* k k)
      (fma
       (/ (* t_m (* t_m t_m)) l)
       (/ 2.0 l)
       (*
        (/ k l)
        (/ (* k (* t_m (+ (* (* t_m t_m) 0.3333333333333333) 1.0))) l)))))
    (*
     2.0
     (/
      (/ (/ l t_m) (* t_m (sin k)))
      (/ (* (+ 2.0 (/ k (/ t_m (/ k t_m)))) (tan k)) (/ l t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-84) {
		tmp = 2.0 / ((k * k) * fma(((t_m * (t_m * t_m)) / l), (2.0 / l), ((k / l) * ((k * (t_m * (((t_m * t_m) * 0.3333333333333333) + 1.0))) / l))));
	} else {
		tmp = 2.0 * (((l / t_m) / (t_m * sin(k))) / (((2.0 + (k / (t_m / (k / t_m)))) * tan(k)) / (l / t_m)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-84)
		tmp = Float64(2.0 / Float64(Float64(k * k) * fma(Float64(Float64(t_m * Float64(t_m * t_m)) / l), Float64(2.0 / l), Float64(Float64(k / l) * Float64(Float64(k * Float64(t_m * Float64(Float64(Float64(t_m * t_m) * 0.3333333333333333) + 1.0))) / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t_m) / Float64(t_m * sin(k))) / Float64(Float64(Float64(2.0 + Float64(k / Float64(t_m / Float64(k / t_m)))) * tan(k)) / Float64(l / t_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-84], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 / l), $MachinePrecision] + N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[(t$95$m * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(2.0 + N[(k / N[(t$95$m / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.00000000000000031e-84

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 55.3%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 2}{\ell \cdot \ell} + \frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot 0.3333333333333333 + 1\right)\right)}{\ell \cdot \ell}\right)}} \]
   6. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \frac{2}{\ell} + \frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}}{\ell \cdot \ell}\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\ell}\right), \color{blue}{\left(\frac{2}{\ell}\right)}, \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \left(\frac{\color{blue}{2}}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \left(\frac{2}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \left(\frac{2}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \color{blue}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \left(\frac{k \cdot \left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \left(\frac{k}{\ell} \cdot \frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\left(\frac{k}{\ell}\right), \left(\frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \left(\frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 66.3%

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

   if 9.00000000000000031e-84 < t

   1. Initial program 70.9%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 89.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 89.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 94.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)}\right), \color{blue}{2}\right) \]

   8. Applied egg-rr 94.8%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot \left(t \cdot t\right)}{\ell}, \frac{2}{\ell}, \frac{k}{\ell} \cdot \frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot 0.3333333333333333 + 1\right)\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t}}{t \cdot \sin k}}{\frac{\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k}{\frac{\ell}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{2 \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t\_m \cdot \left(t\_m \cdot t\_m\right)}{\ell}, \frac{2}{\ell}, \frac{k}{\ell} \cdot \frac{k \cdot \left(t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot 0.3333333333333333 + 1\right)\right)}{\ell}\right)}\\ \end{array} \end{array} \]

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 4.7e-64)
    (/ 2.0 (* (* (* t_m (sin k)) (/ t_m l)) (/ (* 2.0 (* t_m k)) l)))
    (/
     2.0
     (*
      (* k k)
      (fma
       (/ (* t_m (* t_m t_m)) l)
       (/ 2.0 l)
       (*
        (/ k l)
        (/ (* k (* t_m (+ (* (* t_m t_m) 0.3333333333333333) 1.0))) 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 <= 4.7e-64) {
		tmp = 2.0 / (((t_m * sin(k)) * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	} else {
		tmp = 2.0 / ((k * k) * fma(((t_m * (t_m * t_m)) / l), (2.0 / l), ((k / l) * ((k * (t_m * (((t_m * t_m) * 0.3333333333333333) + 1.0))) / 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 <= 4.7e-64)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(t_m / l)) * Float64(Float64(2.0 * Float64(t_m * k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * fma(Float64(Float64(t_m * Float64(t_m * t_m)) / l), Float64(2.0 / l), Float64(Float64(k / l) * Float64(Float64(k * Float64(t_m * Float64(Float64(Float64(t_m * t_m) * 0.3333333333333333) + 1.0))) / 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, 4.7e-64], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 / l), $MachinePrecision] + N[(N[(k / l), $MachinePrecision] * N[(N[(k * N[(t$95$m * N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.6999999999999998e-64

   1. Initial program 61.1%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 78.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 78.0%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 82.4%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(2 \cdot \frac{k \cdot t}{\ell}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2 \cdot \left(k \cdot t\right)}{\color{blue}{\ell}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(k \cdot t\right)\right), \color{blue}{\ell}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \ell\right)\right)\right) \]

   9. Simplified 79.4%

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

   if 4.6999999999999998e-64 < k

   1. Initial program 51.6%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 58.8%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot \left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot 2}{\ell \cdot \ell} + \frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot 0.3333333333333333 + 1\right)\right)}{\ell \cdot \ell}\right)}} \]
   6. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{t \cdot \left(t \cdot t\right)}{\ell} \cdot \frac{2}{\ell} + \frac{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}}{\ell \cdot \ell}\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\left(\frac{t \cdot \left(t \cdot t\right)}{\ell}\right), \color{blue}{\left(\frac{2}{\ell}\right)}, \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \ell\right), \left(\frac{\color{blue}{2}}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \ell\right), \left(\frac{2}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \left(\frac{2}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \color{blue}{\ell}\right), \left(\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \left(\frac{k \cdot \left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)\right)}{\ell \cdot \ell}\right)\right)\right)\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \left(\frac{k}{\ell} \cdot \frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\left(\frac{k}{\ell}\right), \left(\frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \left(\frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)}{\ell}\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \ell\right), \mathsf{/.f64}\left(2, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(k, \ell\right), \mathsf{/.f64}\left(\left(k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot \frac{1}{3} + 1\right)\right)\right), \ell\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 70.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot \left(t \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \mathsf{fma}\left(\frac{t \cdot \left(t \cdot t\right)}{\ell}, \frac{2}{\ell}, \frac{k}{\ell} \cdot \frac{k \cdot \left(t \cdot \left(\left(t \cdot t\right) \cdot 0.3333333333333333 + 1\right)\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m \cdot k}}{\frac{\frac{t\_m}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t\_m}{\frac{k}{t\_m}}}\right) \cdot \left(t\_m \cdot \tan k\right)\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 3.1e-80)
    (/ 2.0 (* (* k k) (/ (/ (* t_m (* k k)) l) l)))
    (/
     (/ 2.0 (* t_m k))
     (/
      (* (/ t_m l) (* (+ 2.0 (/ k (/ t_m (/ k t_m)))) (* t_m (tan 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 (t_m <= 3.1e-80) {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	} else {
		tmp = (2.0 / (t_m * k)) / (((t_m / l) * ((2.0 + (k / (t_m / (k / t_m)))) * (t_m * tan(k)))) / l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.1d-80) then
        tmp = 2.0d0 / ((k * k) * (((t_m * (k * k)) / l) / l))
    else
        tmp = (2.0d0 / (t_m * k)) / (((t_m / l) * ((2.0d0 + (k / (t_m / (k / t_m)))) * (t_m * tan(k)))) / l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.1e-80) {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	} else {
		tmp = (2.0 / (t_m * k)) / (((t_m / l) * ((2.0 + (k / (t_m / (k / t_m)))) * (t_m * Math.tan(k)))) / l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.1e-80:
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l))
	else:
		tmp = (2.0 / (t_m * k)) / (((t_m / l) * ((2.0 + (k / (t_m / (k / t_m)))) * (t_m * math.tan(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 (t_m <= 3.1e-80)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(t_m * Float64(k * k)) / l) / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m * k)) / Float64(Float64(Float64(t_m / l) * Float64(Float64(2.0 + Float64(k / Float64(t_m / Float64(k / t_m)))) * Float64(t_m * tan(k)))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.1e-80)
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	else
		tmp = (2.0 / (t_m * k)) / (((t_m / l) * ((2.0 + (k / (t_m / (k / t_m)))) * (t_m * tan(k)))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.1e-80], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(2.0 + N[(k / N[(t$95$m / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000016e-80

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 55.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{{k}^{2} \cdot t}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\frac{{k}^{2} \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \ell\right), \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \ell\right), \ell\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right), \ell\right)\right)\right) \]

      7. *-lowering-*.f64 64.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right)\right)\right) \]

   8. Simplified 64.7%

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

   if 3.10000000000000016e-80 < t

   1. Initial program 71.4%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 90.8%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 90.8%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 94.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{t \cdot \sin k}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot \sin k\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \sin k\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \left(\frac{t}{\ell} \cdot \frac{t \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)}{\color{blue}{\ell}}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \left(\frac{\frac{t}{\ell} \cdot \left(t \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)}{\color{blue}{\ell}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \tan k\right)\right)\right), \color{blue}{\ell}\right)\right) \]

   8. Applied egg-rr 92.3%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \color{blue}{k}\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(k, t\right)\right)\right)\right)\right)\right), \ell\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 86.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t \cdot k}}{\frac{\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \left(t \cdot \tan k\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{2 \cdot \left(t\_m \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

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 (<= t_m 1.4e-58)
    (/ 2.0 (* (* k k) (/ (/ (* t_m (* k k)) l) l)))
    (/ 2.0 (* (* (* t_m (sin k)) (/ t_m l)) (/ (* 2.0 (* 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 (t_m <= 1.4e-58) {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	} else {
		tmp = 2.0 / (((t_m * sin(k)) * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.4d-58) then
        tmp = 2.0d0 / ((k * k) * (((t_m * (k * k)) / l) / l))
    else
        tmp = 2.0d0 / (((t_m * sin(k)) * (t_m / l)) * ((2.0d0 * (t_m * k)) / l))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.4e-58) {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	} else {
		tmp = 2.0 / (((t_m * Math.sin(k)) * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.4e-58:
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l))
	else:
		tmp = 2.0 / (((t_m * math.sin(k)) * (t_m / l)) * ((2.0 * (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 (t_m <= 1.4e-58)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(t_m * Float64(k * k)) / l) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * sin(k)) * Float64(t_m / l)) * Float64(Float64(2.0 * Float64(t_m * k)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.4e-58)
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	else
		tmp = 2.0 / (((t_m * sin(k)) * (t_m / l)) * ((2.0 * (t_m * k)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.4e-58], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.4e-58

   1. Initial program 53.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 55.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{{k}^{2} \cdot t}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\frac{{k}^{2} \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \ell\right), \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \ell\right), \ell\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right), \ell\right)\right)\right) \]

      7. *-lowering-*.f64 64.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right)\right)\right) \]

   8. Simplified 64.2%

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

   if 1.4e-58 < t

   1. Initial program 70.4%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sin k}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot \sin k\right)}{\ell \cdot \ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot \frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\left(t \cdot t\right) \cdot \sin k\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \sin k\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

      11. sin-lowering-sin.f64 91.5%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right)\right), \ell\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, t\right), 2\right)\right), 1\right)\right)\right) \]

   4. Applied egg-rr 91.5%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t}{\ell} \cdot \frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{t \cdot \left(t \cdot \sin k\right)}{\ell}\right), \color{blue}{\left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{\left(t \cdot \sin k\right) \cdot t}{\ell}\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\color{blue}{\frac{t}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{\color{blue}{t}}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \left(\frac{t}{\ell}\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\color{blue}{\ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(1 + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(1 + 1\right) + \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + {\color{blue}{\left(\frac{k}{t}\right)}}^{2}\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \color{blue}{\frac{k}{t}}\right)\right)\right)\right)\right) \]

      15. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 + \frac{k}{\color{blue}{\frac{t}{\frac{k}{t}}}}\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{t}{\ell} \cdot \left(\left(2 + \frac{k}{\frac{t}{\frac{k}{t}}}\right) \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \color{blue}{\left(2 \cdot \frac{k \cdot t}{\ell}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \left(\frac{2 \cdot \left(k \cdot t\right)}{\color{blue}{\ell}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\left(2 \cdot \left(k \cdot t\right)\right), \color{blue}{\ell}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(k \cdot t\right)\right), \ell\right)\right)\right) \]

      4. *-lowering-*.f64 87.9%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(k, t\right)\right), \ell\right)\right)\right) \]

   9. Simplified 87.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot \left(t \cdot k\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t\_m \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \left(\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{1}{t\_m}}{k}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-73)
    (/ 2.0 (* (* k k) (/ (/ (* t_m (* k k)) l) l)))
    (* (/ l t_m) (* (/ l (* t_m k)) (/ (/ 1.0 t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-73) {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	} else {
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.8d-73) then
        tmp = 2.0d0 / ((k * k) * (((t_m * (k * k)) / l) / l))
    else
        tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0d0 / t_m) / k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.8e-73) {
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	} else {
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.8e-73:
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l))
	else:
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.8e-73)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(Float64(t_m * Float64(k * k)) / l) / l)));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(1.0 / t_m) / k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.8e-73)
		tmp = 2.0 / ((k * k) * (((t_m * (k * k)) / l) / l));
	else
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-73], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000012e-73

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 55.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{{k}^{2} \cdot t}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\frac{{k}^{2} \cdot t}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\frac{{k}^{2} \cdot t}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \ell\right), \ell\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \ell\right), \ell\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \ell\right), \ell\right)\right)\right) \]

      7. *-lowering-*.f64 65.0%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \ell\right), \ell\right)\right)\right) \]

   8. Simplified 65.0%

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

   if 2.80000000000000012e-73 < t

   1. Initial program 70.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 60.0%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 70.5%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell \cdot \frac{1}{t}}{\color{blue}{t} \cdot \left(k \cdot k\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell \cdot \frac{1}{t}}{\left(t \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{t \cdot k} \cdot \color{blue}{\frac{\frac{1}{t}}{k}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{\frac{1}{t}}{k}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\frac{1}{t}}}{k}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\frac{1}{\color{blue}{t}}}{k}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{t}\right), \color{blue}{k}\right)\right)\right) \]

      9. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), k\right)\right)\right) \]

   9. Applied egg-rr 85.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\ell}{t \cdot k} \cdot \frac{\frac{1}{t}}{k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t\_m \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \left(\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{1}{t\_m}}{k}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-73)
    (/ 2.0 (* (* k k) (/ (* t_m (* k k)) (* l l))))
    (* (/ l t_m) (* (/ l (* t_m k)) (/ (/ 1.0 t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-73) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else {
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.6d-73) then
        tmp = 2.0d0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
    else
        tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0d0 / t_m) / k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-73) {
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	} else {
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.6e-73:
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)))
	else:
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.6e-73)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t_m * Float64(k * k)) / Float64(l * l))));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(1.0 / t_m) / k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.6e-73)
		tmp = 2.0 / ((k * k) * ((t_m * (k * k)) / (l * l)));
	else
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-73], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.6000000000000001e-73

   1. Initial program 52.5%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2}\right), \color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\color{blue}{2 \cdot \frac{{t}^{3}}{{\ell}^{2}}} + \frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot {t}^{3}}{{\ell}^{2}}\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {t}^{3}\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({t}^{3} \cdot 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({t}^{3}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{\color{blue}{{k}^{2}} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{\color{blue}{k}}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left({\ell}^{2}\right)\right), \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}{{\ell}^{2}}\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \left(\ell \cdot \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{{k}^{2} \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)}}{{\ell}^{2}}\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), 2\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right) \]

   5. Simplified 55.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right)}{{\ell}^{2}}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right)\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left(\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2} \cdot t\right), \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \left(1 + \frac{1}{3} \cdot {t}^{2}\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \left(\frac{1}{3} \cdot {t}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \left({t}^{2} \cdot \frac{1}{3}\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({t}^{2}\right), \frac{1}{3}\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot t\right), \frac{1}{3}\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left({\ell}^{2}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 59.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \frac{1}{3}\right)\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   8. Simplified 59.2%

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

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \color{blue}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\left({k}^{2} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), t\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), t\right), \left({\ell}^{2}\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \left({\ell}^{2}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 59.3%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 59.3%

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

   if 2.6000000000000001e-73 < t

   1. Initial program 70.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 60.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 60.0%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 70.5%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell \cdot \frac{1}{t}}{\color{blue}{t} \cdot \left(k \cdot k\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell \cdot \frac{1}{t}}{\left(t \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{t \cdot k} \cdot \color{blue}{\frac{\frac{1}{t}}{k}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{\frac{1}{t}}{k}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\frac{1}{t}}}{k}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\frac{1}{\color{blue}{t}}}{k}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{t}\right), \color{blue}{k}\right)\right)\right) \]

      9. /-lowering-/.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), k\right)\right)\right) \]

   9. Applied egg-rr 85.9%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\ell}{t \cdot k} \cdot \frac{\frac{1}{t}}{k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}{\ell}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \left(\frac{\ell}{t\_m \cdot k} \cdot \frac{\frac{1}{t\_m}}{k}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.9e-90)
    (/ (/ l (/ (* t_m (* t_m (* k k))) l)) t_m)
    (* (/ l t_m) (* (/ l (* t_m k)) (/ (/ 1.0 t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e-90) {
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m;
	} else {
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.9d-90) then
        tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m
    else
        tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0d0 / t_m) / k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.9e-90) {
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m;
	} else {
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.9e-90:
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m
	else:
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.9e-90)
		tmp = Float64(Float64(l / Float64(Float64(t_m * Float64(t_m * Float64(k * k))) / l)) / t_m);
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(t_m * k)) * Float64(Float64(1.0 / t_m) / k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.9e-90)
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m;
	else
		tmp = (l / t_m) * ((l / (t_m * k)) * ((1.0 / t_m) / k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.9e-90], N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.89999999999999983e-90

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 50.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 50.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 66.0%

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

      1. associate-*l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\right), t\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\right), t\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right), t\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right), t\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right), t\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \ell\right)\right), t\right) \]

      9. *-lowering-*.f64 67.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right)\right), t\right) \]

   9. Applied egg-rr 67.2%

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

   if 2.89999999999999983e-90 < t

   1. Initial program 69.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 58.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 58.9%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 68.3%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell \cdot \frac{1}{t}}{\color{blue}{t} \cdot \left(k \cdot k\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell \cdot \frac{1}{t}}{\left(t \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\ell}{t \cdot k} \cdot \color{blue}{\frac{\frac{1}{t}}{k}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\left(\frac{\ell}{t \cdot k}\right), \color{blue}{\left(\frac{\frac{1}{t}}{k}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot k\right)\right), \left(\frac{\color{blue}{\frac{1}{t}}}{k}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \left(\frac{\frac{1}{\color{blue}{t}}}{k}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{t}\right), \color{blue}{k}\right)\right)\right) \]

      9. /-lowering-/.f64 81.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, k\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), k\right)\right)\right) \]

   9. Applied egg-rr 81.9%

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

Alternative 11: 72.5% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}{\ell}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-90)
    (/ (/ l (/ (* t_m (* t_m (* k k))) l)) t_m)
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-90) {
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m;
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.5d-90) then
        tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m
    else
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.5e-90) {
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m;
	} else {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.5e-90:
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m
	else:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.5e-90)
		tmp = Float64(Float64(l / Float64(Float64(t_m * Float64(t_m * Float64(k * k))) / l)) / t_m);
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.5e-90)
		tmp = (l / ((t_m * (t_m * (k * k))) / l)) / t_m;
	else
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-90], N[(N[(l / N[(N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.4999999999999999e-90

   1. Initial program 52.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 50.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 50.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 66.0%

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

      1. associate-*l/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \frac{1}{\frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\right), t\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\right), t\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}\right)\right), t\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right), t\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right), \ell\right)\right), t\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right), \ell\right)\right), t\right) \]

      9. *-lowering-*.f64 67.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right), \ell\right)\right), t\right) \]

   9. Applied egg-rr 67.2%

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

   if 3.4999999999999999e-90 < t

   1. Initial program 69.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 58.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 58.9%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 68.3%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{k}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{k}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right)\right) \]

      7. *-lowering-*.f64 81.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right)\right) \]

   9. Applied egg-rr 81.9%

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

Alternative 12: 70.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{+180}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{t\_m \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

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 1e+180)
    (* (/ l t_m) (/ (/ (/ l t_m) (* t_m k)) 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 <= 1e+180) {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / 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 <= 1d+180) then
        tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / 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 <= 1e+180) {
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / 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 <= 1e+180:
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / 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 <= 1e+180)
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(t_m * k)) / k));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e+180)
		tmp = (l / t_m) * (((l / t_m) / (t_m * k)) / 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, 1e+180], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1e180

   1. Initial program 58.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 52.9%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 65.1%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\left(t \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\frac{\ell}{t}}{t \cdot k}}{\color{blue}{k}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot k}\right), \color{blue}{k}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(t \cdot k\right)\right), k\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(t \cdot k\right)\right), k\right)\right) \]

      7. *-lowering-*.f64 75.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(t, k\right)\right), k\right)\right) \]

   9. Applied egg-rr 75.3%

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

   if 1e180 < k

   1. Initial program 55.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 55.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 75.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      8. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

   9. Applied egg-rr 76.0%

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

Alternative 13: 70.2% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\frac{\ell}{t\_m}}{k}}{t\_m \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\frac{\ell}{t\_m}}{k \cdot k}}{t\_m}\\ \end{array} \end{array} \]

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 5.8e-159)
    (* l (/ (/ (/ (/ l t_m) k) (* t_m k)) t_m))
    (* (/ l t_m) (/ (/ (/ l t_m) (* k k)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-159) {
		tmp = l * ((((l / t_m) / k) / (t_m * k)) / t_m);
	} else {
		tmp = (l / t_m) * (((l / t_m) / (k * k)) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-159) then
        tmp = l * ((((l / t_m) / k) / (t_m * k)) / t_m)
    else
        tmp = (l / t_m) * (((l / t_m) / (k * k)) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-159) {
		tmp = l * ((((l / t_m) / k) / (t_m * k)) / t_m);
	} else {
		tmp = (l / t_m) * (((l / t_m) / (k * k)) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.8e-159:
		tmp = l * ((((l / t_m) / k) / (t_m * k)) / t_m)
	else:
		tmp = (l / t_m) * (((l / t_m) / (k * k)) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.8e-159)
		tmp = Float64(l * Float64(Float64(Float64(Float64(l / t_m) / k) / Float64(t_m * k)) / t_m));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(Float64(l / t_m) / Float64(k * k)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.8e-159)
		tmp = l * ((((l / t_m) / k) / (t_m * k)) / t_m);
	else
		tmp = (l / t_m) * (((l / t_m) / (k * k)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.8e-159], N[(l * N[(N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(N[(l / t$95$m), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.79999999999999981e-159

   1. Initial program 60.9%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 53.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 53.0%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 65.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 65.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      8. *-lowering-*.f64 66.0%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

   9. Applied egg-rr 66.0%

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

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), t\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot t}\right), t\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}\right), t\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k}}{k \cdot t}\right), t\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k}}{t \cdot k}\right), t\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \left(t \cdot k\right)\right), t\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(t \cdot k\right)\right), t\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(t \cdot k\right)\right), t\right)\right) \]

       9. *-lowering-*.f64 78.6%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{*.f64}\left(t, k\right)\right), t\right)\right) \]

   11. Applied egg-rr 78.6%

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

   if 5.79999999999999981e-159 < k

   1. Initial program 53.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 53.6%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 68.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 68.0%

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

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\color{blue}{t \cdot \left(k \cdot k\right)}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot \color{blue}{t}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\frac{\frac{\ell}{t}}{k \cdot k}}{\color{blue}{t}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot k}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(k \cdot k\right)\right), t\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 70.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), t\right)\right) \]

   9. Applied egg-rr 70.7%

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

Alternative 14: 69.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{+180}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\frac{\ell}{t\_m}}{k}}{t\_m \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

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 1e+180)
    (* l (/ (/ (/ (/ l t_m) 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 <= 1e+180) {
		tmp = l * ((((l / t_m) / 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 <= 1d+180) then
        tmp = l * ((((l / t_m) / 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 <= 1e+180) {
		tmp = l * ((((l / t_m) / 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 <= 1e+180:
		tmp = l * ((((l / t_m) / 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 <= 1e+180)
		tmp = Float64(l * Float64(Float64(Float64(Float64(l / t_m) / k) / Float64(t_m * k)) / t_m));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e+180)
		tmp = l * ((((l / t_m) / 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, 1e+180], N[(l * N[(N[(N[(N[(l / t$95$m), $MachinePrecision] / k), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1e180

   1. Initial program 58.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 52.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 52.9%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 65.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      8. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

   9. Applied egg-rr 65.2%

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

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot k\right)}\right), t\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot t}\right), t\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k \cdot \left(k \cdot t\right)}\right), t\right)\right) \]

       4. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k}}{k \cdot t}\right), t\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\frac{\frac{\ell}{t}}{k}}{t \cdot k}\right), t\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{k}\right), \left(t \cdot k\right)\right), t\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), k\right), \left(t \cdot k\right)\right), t\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \left(t \cdot k\right)\right), t\right)\right) \]

       9. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), k\right), \mathsf{*.f64}\left(t, k\right)\right), t\right)\right) \]

   11. Applied egg-rr 74.4%

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

   if 1e180 < k

   1. Initial program 55.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 55.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 55.1%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 75.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 75.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      8. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

   9. Applied egg-rr 76.0%

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

Alternative 15: 67.2% accurate, 23.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-234}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot \left(k \cdot \left(t\_m \cdot k\right)\right)}}{t\_m}\\ \end{array} \end{array} \]

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 2.3e-234)
    (* l (/ (/ l (* t_m (* t_m (* k k)))) t_m))
    (* l (/ (/ l (* t_m (* k (* t_m k)))) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-234) {
		tmp = l * ((l / (t_m * (t_m * (k * k)))) / t_m);
	} else {
		tmp = l * ((l / (t_m * (k * (t_m * 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 <= 2.3d-234) then
        tmp = l * ((l / (t_m * (t_m * (k * k)))) / t_m)
    else
        tmp = l * ((l / (t_m * (k * (t_m * 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 <= 2.3e-234) {
		tmp = l * ((l / (t_m * (t_m * (k * k)))) / t_m);
	} else {
		tmp = l * ((l / (t_m * (k * (t_m * 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 <= 2.3e-234:
		tmp = l * ((l / (t_m * (t_m * (k * k)))) / t_m)
	else:
		tmp = l * ((l / (t_m * (k * (t_m * 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 <= 2.3e-234)
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * Float64(k * Float64(t_m * 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 <= 2.3e-234)
		tmp = l * ((l / (t_m * (t_m * (k * k)))) / t_m);
	else
		tmp = l * ((l / (t_m * (k * (t_m * 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, 2.3e-234], N[(l * N[(N[(l / N[(t$95$m * N[(t$95$m * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * N[(k * N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2999999999999999e-234

   1. Initial program 57.0%

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

   3. Taylor expanded in k around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 54.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 54.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 69.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      8. *-lowering-*.f64 69.8%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

   9. Applied egg-rr 69.8%

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

   if 2.2999999999999999e-234 < t

   1. Initial program 58.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

      12. *-lowering-*.f64 51.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   5. Simplified 51.6%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 62.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 62.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

      8. *-lowering-*.f64 62.9%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

   9. Applied egg-rr 62.9%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(\left(t \cdot k\right) \cdot k\right)\right)\right), t\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(t \cdot k\right), k\right)\right)\right), t\right)\right) \]

       3. *-lowering-*.f64 69.7%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, k\right), k\right)\right)\right), t\right)\right) \]

   11. Applied egg-rr 69.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-234}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(k \cdot k\right)\right)}}{t\_m}\right) \end{array} \]

FPCore

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

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * ((l / (t_m * (t_m * (k * k)))) / t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (l * ((l / (t_m * (t_m * (k * k)))) / t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (l * ((l / (t_m * (t_m * (k * k)))) / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * ((l / (t_m * (t_m * (k * k)))) / t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(Float64(l / Float64(t_m * Float64(t_m * Float64(k * k)))) / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (l * ((l / (t_m * (t_m * (k * k)))) / t_m));
end

Wolfram

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

Derivation

1. Initial program 57.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
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. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left({k}^{2} \cdot {t}^{3}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{{k}^{2}} \cdot {t}^{3}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \color{blue}{{k}^{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({t}^{3}\right), \color{blue}{\left({k}^{2}\right)}\right)\right) \]

   6. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot \left(t \cdot t\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(t \cdot {t}^{2}\right), \left({k}^{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), \left({\color{blue}{k}}^{2}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), \left({k}^{2}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left({k}^{2}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \left(k \cdot \color{blue}{k}\right)\right)\right) \]

   12. *-lowering-*.f64 53.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

5. Simplified 53.2%

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

   1. associate-*l* N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\ell}}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \left(t \cdot \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \color{blue}{\left(t \cdot \left(k \cdot k\right)\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \color{blue}{\left(k \cdot k\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 66.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 66.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}}{t}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\frac{\ell}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right), \color{blue}{t}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \left(k \cdot k\right)\right)\right)\right), t\right)\right) \]

   8. *-lowering-*.f64 66.9%

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(k, k\right)\right)\right)\right), t\right)\right) \]

9. Applied egg-rr 66.9%

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

Reproduce

herbie shell --seed 2024192 
(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))))