Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 4.0

Time: 46.4s

Precision: binary64

Cost: 21004

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t_1 \cdot \frac{\sin k}{\frac{1}{t}}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+173}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t_1}{\frac{\ell}{k}} \cdot \left(\sin k \cdot t\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (sin k) l)))
   (if (<= (* l l) 5e-11)
     (* (cos k) (/ (/ 2.0 k) (/ (* t_1 (/ (sin k) (/ 1.0 t))) (/ l k))))
     (if (<= (* l l) 1e+173)
       (* (cos k) (/ (/ 2.0 k) (* (/ t_1 (/ l k)) (* (sin k) t))))
       (if (<= (* l l) 5e+198)
         (* 2.0 (* (/ (cos k) (* k k)) (/ (* l l) (* t (pow (sin k) 2.0)))))
         (/ -2.0 (* t (* (/ k l) (* (/ k l) (* (sin k) (- (tan k))))))))))))

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));
}

↓

double code(double t, double l, double k) {
	double t_1 = sin(k) / l;
	double tmp;
	if ((l * l) <= 5e-11) {
		tmp = cos(k) * ((2.0 / k) / ((t_1 * (sin(k) / (1.0 / t))) / (l / k)));
	} else if ((l * l) <= 1e+173) {
		tmp = cos(k) * ((2.0 / k) / ((t_1 / (l / k)) * (sin(k) * t)));
	} else if ((l * l) <= 5e+198) {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = -2.0 / (t * ((k / l) * ((k / l) * (sin(k) * -tan(k)))));
	}
	return tmp;
}

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) / l
    if ((l * l) <= 5d-11) then
        tmp = cos(k) * ((2.0d0 / k) / ((t_1 * (sin(k) / (1.0d0 / t))) / (l / k)))
    else if ((l * l) <= 1d+173) then
        tmp = cos(k) * ((2.0d0 / k) / ((t_1 / (l / k)) * (sin(k) * t)))
    else if ((l * l) <= 5d+198) then
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l * l) / (t * (sin(k) ** 2.0d0))))
    else
        tmp = (-2.0d0) / (t * ((k / l) * ((k / l) * (sin(k) * -tan(k)))))
    end if
    code = tmp
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));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) / l;
	double tmp;
	if ((l * l) <= 5e-11) {
		tmp = Math.cos(k) * ((2.0 / k) / ((t_1 * (Math.sin(k) / (1.0 / t))) / (l / k)));
	} else if ((l * l) <= 1e+173) {
		tmp = Math.cos(k) * ((2.0 / k) / ((t_1 / (l / k)) * (Math.sin(k) * t)));
	} else if ((l * l) <= 5e+198) {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l * l) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = -2.0 / (t * ((k / l) * ((k / l) * (Math.sin(k) * -Math.tan(k)))));
	}
	return tmp;
}

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))

↓

def code(t, l, k):
	t_1 = math.sin(k) / l
	tmp = 0
	if (l * l) <= 5e-11:
		tmp = math.cos(k) * ((2.0 / k) / ((t_1 * (math.sin(k) / (1.0 / t))) / (l / k)))
	elif (l * l) <= 1e+173:
		tmp = math.cos(k) * ((2.0 / k) / ((t_1 / (l / k)) * (math.sin(k) * t)))
	elif (l * l) <= 5e+198:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l * l) / (t * math.pow(math.sin(k), 2.0))))
	else:
		tmp = -2.0 / (t * ((k / l) * ((k / l) * (math.sin(k) * -math.tan(k)))))
	return tmp

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

↓

function code(t, l, k)
	t_1 = Float64(sin(k) / l)
	tmp = 0.0
	if (Float64(l * l) <= 5e-11)
		tmp = Float64(cos(k) * Float64(Float64(2.0 / k) / Float64(Float64(t_1 * Float64(sin(k) / Float64(1.0 / t))) / Float64(l / k))));
	elseif (Float64(l * l) <= 1e+173)
		tmp = Float64(cos(k) * Float64(Float64(2.0 / k) / Float64(Float64(t_1 / Float64(l / k)) * Float64(sin(k) * t))));
	elseif (Float64(l * l) <= 5e+198)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l * l) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(-2.0 / Float64(t * Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(sin(k) * Float64(-tan(k)))))));
	end
	return tmp
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

↓

function tmp_2 = code(t, l, k)
	t_1 = sin(k) / l;
	tmp = 0.0;
	if ((l * l) <= 5e-11)
		tmp = cos(k) * ((2.0 / k) / ((t_1 * (sin(k) / (1.0 / t))) / (l / k)));
	elseif ((l * l) <= 1e+173)
		tmp = cos(k) * ((2.0 / k) / ((t_1 / (l / k)) * (sin(k) * t)));
	elseif ((l * l) <= 5e+198)
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * (sin(k) ^ 2.0))));
	else
		tmp = -2.0 / (t * ((k / l) * ((k / l) * (sin(k) * -tan(k)))));
	end
	tmp_2 = tmp;
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]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e-11], N[(N[Cos[k], $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+173], N[(N[Cos[k], $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[(t$95$1 / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+198], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(t * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * (-N[Tan[k], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 l l) < 5.00000000000000018e-11

   1. Initial program 43.7

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

   2. Simplified 34.5

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1))))): 43 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0)))): 51 points increase in error, 0 points decrease in error
      (/.f64 2 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (+.f64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite=> mul0-rgt_binary64 0))): 0 points increase in error, 51 points decrease in error
      (/.f64 2 (Rewrite=> +-rgt-identity_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 13.2

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

   4. Simplified 7.3

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]
      Proof
      (*.f64 (/.f64 (*.f64 k k) (cos.f64 k)) (*.f64 (/.f64 t l) (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (cos.f64 k)) (*.f64 (/.f64 t l) (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (pow.f64 k 2) (cos.f64 k)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) (*.f64 l l)))): 26 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 (pow.f64 k 2) (cos.f64 k)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (*.f64 l l)))): 20 points increase in error, 13 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in k around inf 13.1

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

   6. Simplified 5.6

      \[\leadsto \color{blue}{\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}{\frac{\ell}{k}}}} \]
      Proof
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (/.f64 (pow.f64 (sin.f64 k) 2) (/.f64 l t)) (/.f64 l k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) l)) (/.f64 l k)))): 19 points increase in error, 15 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t (pow.f64 (sin.f64 k) 2))) l) (/.f64 l k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2))) (/.f64 l k)))): 20 points increase in error, 18 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2)) k) l)))): 11 points increase in error, 14 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2)) l) k)))): 43 points increase in error, 15 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (*.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))) k))): 6 points increase in error, 12 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (Rewrite=> *-commutative_binary64 (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 k) (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 k (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))))): 20 points increase in error, 14 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 (*.f64 k (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))) (cos.f64 k))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 2 (cos.f64 k)) (*.f64 k (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2))))))): 8 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 2 (cos.f64 k)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 k k) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))): 28 points increase in error, 5 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (*.f64 2 (cos.f64 k)) (*.f64 k k)) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2))))): 9 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (cos.f64 k) (*.f64 k k)))) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2)) l)))): 9 points increase in error, 6 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) l)) l))): 9 points increase in error, 12 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) (*.f64 l l))))): 29 points increase in error, 4 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (*.f64 l l)) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 7 points increase in error, 21 points decrease in error
      (*.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (*.f64 l l) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 23 points increase in error, 11 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.f64 (*.f64 k k) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 29 points increase in error, 22 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.f64 (*.f64 k k) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.f64 (*.f64 k k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 2.5

      \[\leadsto \cos k \cdot \frac{\frac{2}{k}}{\frac{\color{blue}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\frac{1}{t}}}}{\frac{\ell}{k}}} \]

   if 5.00000000000000018e-11 < (*.f64 l l) < 1e173

   1. Initial program 45.6

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

   2. Simplified 37.3

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1))))): 43 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0)))): 51 points increase in error, 0 points decrease in error
      (/.f64 2 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (+.f64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite=> mul0-rgt_binary64 0))): 0 points increase in error, 51 points decrease in error
      (/.f64 2 (Rewrite=> +-rgt-identity_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 14.6

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

   4. Simplified 14.8

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]
      Proof
      (*.f64 (/.f64 (*.f64 k k) (cos.f64 k)) (*.f64 (/.f64 t l) (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (cos.f64 k)) (*.f64 (/.f64 t l) (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (pow.f64 k 2) (cos.f64 k)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) (*.f64 l l)))): 26 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 (pow.f64 k 2) (cos.f64 k)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (*.f64 l l)))): 20 points increase in error, 13 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in k around inf 14.6

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

   6. Simplified 4.7

      \[\leadsto \color{blue}{\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}{\frac{\ell}{k}}}} \]
      Proof
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (/.f64 (pow.f64 (sin.f64 k) 2) (/.f64 l t)) (/.f64 l k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) l)) (/.f64 l k)))): 19 points increase in error, 15 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 t (pow.f64 (sin.f64 k) 2))) l) (/.f64 l k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2))) (/.f64 l k)))): 20 points increase in error, 18 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2)) k) l)))): 11 points increase in error, 14 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2)) l) k)))): 43 points increase in error, 15 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (*.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))) k))): 6 points increase in error, 12 points decrease in error
      (*.f64 (cos.f64 k) (/.f64 (/.f64 2 k) (Rewrite=> *-commutative_binary64 (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 k) (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 k (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))))): 20 points increase in error, 14 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 (*.f64 k (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))) (cos.f64 k))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 2 (cos.f64 k)) (*.f64 k (*.f64 k (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2))))))): 8 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 2 (cos.f64 k)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 k k) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))): 28 points increase in error, 5 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (*.f64 2 (cos.f64 k)) (*.f64 k k)) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2))))): 9 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (cos.f64 k) (*.f64 k k)))) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (/.f64 t l) (/.f64 l (pow.f64 (sin.f64 k) 2)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 t l) (pow.f64 (sin.f64 k) 2)) l)))): 9 points increase in error, 6 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) l)) l))): 9 points increase in error, 12 points decrease in error
      (*.f64 2 (/.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) (*.f64 l l))))): 29 points increase in error, 4 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (*.f64 l l)) (*.f64 t (pow.f64 (sin.f64 k) 2))))): 7 points increase in error, 21 points decrease in error
      (*.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (*.f64 l l) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 23 points increase in error, 11 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.f64 (*.f64 k k) (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 29 points increase in error, 22 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.f64 (*.f64 k k) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.f64 (*.f64 k k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 3.8

      \[\leadsto \cos k \cdot \frac{\frac{2}{k}}{\frac{\color{blue}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\frac{1}{t}}}}{\frac{\ell}{k}}} \]

   8. Applied egg-rr 1.4

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

   if 1e173 < (*.f64 l l) < 5.00000000000000049e198

   1. Initial program 47.7

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

   2. Simplified 41.0

      \[\leadsto \color{blue}{\frac{2}{\left({t}^{3} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)} \]
      Proof
      (*.f64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))) (pow.f64 (/.f64 k t) 2))) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k))) (pow.f64 (/.f64 k t) 2))) (*.f64 l l)): 3 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))) (*.f64 l l)): 40 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k))))) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (*.f64 (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k))) (*.f64 l l)))): 2 points increase in error, 3 points decrease in error
      (/.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (/.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)) (*.f64 l l))))): 3 points increase in error, 2 points decrease in error
      (/.f64 2 (*.f64 (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 l l)) (tan.f64 k))))): 1 points increase in error, 4 points decrease in error
      (/.f64 2 (*.f64 (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k))) (tan.f64 k)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 22.7

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

   4. Simplified 25.0

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right)} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (/.f64 (*.f64 l l) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (*.f64 l l) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 29 points increase in error, 22 points decrease in error
      (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error

   if 5.00000000000000049e198 < (*.f64 l l)

   1. Initial program 58.4

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

   2. Simplified 56.4

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1))))): 43 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0)))): 51 points increase in error, 0 points decrease in error
      (/.f64 2 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) 0))): 0 points increase in error, 1 points decrease in error
      (/.f64 2 (+.f64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite=> mul0-rgt_binary64 0))): 0 points increase in error, 51 points decrease in error
      (/.f64 2 (Rewrite=> +-rgt-identity_binary64 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in t around 0 51.9

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

   4. Simplified 40.7

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]
      Proof
      (*.f64 (/.f64 (*.f64 k k) (cos.f64 k)) (*.f64 (/.f64 t l) (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (cos.f64 k)) (*.f64 (/.f64 t l) (/.f64 (pow.f64 (sin.f64 k) 2) l))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (pow.f64 k 2) (cos.f64 k)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (pow.f64 (sin.f64 k) 2)) (*.f64 l l)))): 26 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 (pow.f64 k 2) (cos.f64 k)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (*.f64 l l)))): 20 points increase in error, 13 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in k around inf 51.9

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

   6. Simplified 17.9

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\ell}{k}}}{\frac{\ell}{t}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 k (/.f64 l k)) (/.f64 l t)) (/.f64 (pow.f64 (sin.f64 k) 2) (cos.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 k k) l)) (/.f64 l t)) (/.f64 (pow.f64 (sin.f64 k) 2) (cos.f64 k))): 31 points increase in error, 14 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 k k) (*.f64 l (/.f64 l t)))) (/.f64 (pow.f64 (sin.f64 k) 2) (cos.f64 k))): 29 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 (*.f64 k k) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 l l) t))) (/.f64 (pow.f64 (sin.f64 k) 2) (cos.f64 k))): 17 points increase in error, 10 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 k k) t) (*.f64 l l))) (/.f64 (pow.f64 (sin.f64 k) 2) (cos.f64 k))): 15 points increase in error, 20 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (*.f64 k k) t) (pow.f64 (sin.f64 k) 2)) (*.f64 (*.f64 l l) (cos.f64 k)))): 14 points increase in error, 11 points decrease in error
      (/.f64 (*.f64 (*.f64 (*.f64 k k) t) (pow.f64 (sin.f64 k) 2)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (*.f64 l l)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 k k) (*.f64 t (pow.f64 (sin.f64 k) 2)))) (*.f64 (cos.f64 k) (*.f64 l l))): 3 points increase in error, 3 points decrease in error
      (/.f64 (*.f64 (*.f64 k k) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (cos.f64 k) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (*.f64 l l))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 17.8

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

   8. Simplified 8.0

      \[\leadsto \color{blue}{\frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}} \]
      Proof
      (/.f64 -2 (*.f64 t (*.f64 (/.f64 k l) (*.f64 (/.f64 k l) (*.f64 (sin.f64 k) (neg.f64 (tan.f64 k))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (*.f64 -2 1)) (*.f64 t (*.f64 (/.f64 k l) (*.f64 (/.f64 k l) (*.f64 (sin.f64 k) (neg.f64 (tan.f64 k))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -2 1) (*.f64 t (*.f64 (/.f64 k l) (*.f64 (/.f64 k l) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 k) (tan.f64 k)))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -2 1) (*.f64 t (*.f64 (/.f64 k l) (*.f64 (/.f64 k l) (neg.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (sin.f64 k) 1)) (tan.f64 k))))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -2 1) (*.f64 t (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 k l) (/.f64 k l)) (neg.f64 (*.f64 (/.f64 (sin.f64 k) 1) (tan.f64 k))))))): 18 points increase in error, 10 points decrease in error
      (/.f64 (*.f64 -2 1) (*.f64 t (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 k (/.f64 l (/.f64 k l)))) (neg.f64 (*.f64 (/.f64 (sin.f64 k) 1) (tan.f64 k)))))): 20 points increase in error, 13 points decrease in error
      (/.f64 (*.f64 -2 1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t (/.f64 k (/.f64 l (/.f64 k l)))) (neg.f64 (*.f64 (/.f64 (sin.f64 k) 1) (tan.f64 k)))))): 7 points increase in error, 10 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -2 (/.f64 1 (*.f64 (*.f64 t (/.f64 k (/.f64 l (/.f64 k l)))) (neg.f64 (*.f64 (/.f64 (sin.f64 k) 1) (tan.f64 k))))))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\frac{1}{t}}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+173}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{\sin k}{\ell}}{\frac{\ell}{k}} \cdot \left(\sin k \cdot t\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	20884

\[\begin{array}{l} t_1 := \frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ t_2 := t \cdot {\sin k}^{2}\\ t_3 := \cos k \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t_2}\right)\right)\\ \mathbf{if}\;\ell \leq -1.86 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.32 \cdot 10^{-108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{-222}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{\sin k}{\ell} \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_2}{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\frac{\sin k \cdot \tan k}{\frac{2}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	4.8
Cost	20816

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \cos k \cdot \left(\frac{\frac{2}{k}}{t_1} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot t\right)}\right)\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{+184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;\cos k \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t \cdot t_3}\right)\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t_1 \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+158}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{t_3 \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	5.3
Cost	20816

\[\begin{array}{l} t_1 := \sin k \cdot t\\ t_2 := \frac{\sin k}{\ell}\\ t_3 := \cos k \cdot \left(\frac{\frac{2}{k}}{t_2} \cdot \frac{\ell}{k \cdot t_1}\right)\\ \mathbf{if}\;k \leq -2 \cdot 10^{+184}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -6.1 \cdot 10^{-62}:\\ \;\;\;\;\cos k \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t \cdot {\sin k}^{2}}\right)\right)\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t_2}{\frac{\ell}{k}} \cdot t_1}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+158}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\sin k \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	4.3
Cost	20816

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \cos k \cdot \left(\frac{\frac{2}{k}}{t_1} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot t\right)}\right)\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2.1 \cdot 10^{+185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;\cos k \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t \cdot t_3}\right)\right)\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-56}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t_1 \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+158}:\\ \;\;\;\;\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\frac{-2}{k}}{t_3}}{\frac{-t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	5.2
Cost	20752

\[\begin{array}{l} t_1 := \cos k \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t \cdot {\sin k}^{2}}\right)\right)\\ t_2 := \frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-226}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{\sin k}{\ell} \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	5.6
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{\sin k}{\ell} \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{\sin k \cdot \tan k} \cdot \left(\frac{\ell}{k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	5.7
Cost	14344

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{\frac{\sin k}{\ell} \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	6.6
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \left(\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	13.4
Cost	14024

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\ell \cdot \frac{2}{k \cdot k}}{\sin k \cdot \tan k}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-64}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	9.1
Cost	14024

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{2}{k}}{\frac{k}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-55}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	6.4
Cost	14024

\[\begin{array}{l} t_1 := \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{if}\;k \leq -6.2 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-10}:\\ \;\;\;\;\cos k \cdot \frac{\frac{2}{k}}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	24.2
Cost	13956

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-249}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{k \cdot \left(\frac{k}{\ell} \cdot \frac{k}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 13
Error	25.1
Cost	7488

\[\cos k \cdot \frac{\frac{2}{k}}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\frac{\ell}{k}}} \]

Alternative 14
Error	26.1
Cost	960

\[\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell \cdot 2}{k \cdot k} \]

Reproduce

herbie shell --seed 2022321 
(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))))