Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 4.3

Time: 38.8s

Precision: binary64

Cost: 26892

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := \frac{\cos k}{t}\\ t_3 := t_2 \cdot \ell\\ \mathbf{if}\;t \leq -1 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t_1} \cdot \frac{\ell \cdot \ell}{t_1}\right)\\ \mathbf{elif}\;t \leq 0:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{1}{{\left(\sqrt{t} \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\right)\\ \end{array} \]

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 (* k (sin k))) (t_2 (/ (cos k) t)) (t_3 (* t_2 l)))
   (if (<= t -1e-265)
     (* 2.0 (* (/ t_3 t_1) (/ l t_1)))
     (if (<= t -1e-300)
       (* 2.0 (* (/ t_2 t_1) (/ (* l l) t_1)))
       (if (<= t 0.0)
         (* 2.0 (* (/ t_3 (* k k)) (/ l (pow (sin k) 2.0))))
         (*
          2.0
          (* (cos k) (/ 1.0 (pow (* (sqrt t) (* k (/ (sin k) l))) 2.0)))))))))

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));
}

↓

double code(double t, double l, double k) {
	double t_1 = k * sin(k);
	double t_2 = cos(k) / t;
	double t_3 = t_2 * l;
	double tmp;
	if (t <= -1e-265) {
		tmp = 2.0 * ((t_3 / t_1) * (l / t_1));
	} else if (t <= -1e-300) {
		tmp = 2.0 * ((t_2 / t_1) * ((l * l) / t_1));
	} else if (t <= 0.0) {
		tmp = 2.0 * ((t_3 / (k * k)) * (l / pow(sin(k), 2.0)));
	} else {
		tmp = 2.0 * (cos(k) * (1.0 / pow((sqrt(t) * (k * (sin(k) / l))), 2.0)));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = k * sin(k)
    t_2 = cos(k) / t
    t_3 = t_2 * l
    if (t <= (-1d-265)) then
        tmp = 2.0d0 * ((t_3 / t_1) * (l / t_1))
    else if (t <= (-1d-300)) then
        tmp = 2.0d0 * ((t_2 / t_1) * ((l * l) / t_1))
    else if (t <= 0.0d0) then
        tmp = 2.0d0 * ((t_3 / (k * k)) * (l / (sin(k) ** 2.0d0)))
    else
        tmp = 2.0d0 * (cos(k) * (1.0d0 / ((sqrt(t) * (k * (sin(k) / l))) ** 2.0d0)))
    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 = k * Math.sin(k);
	double t_2 = Math.cos(k) / t;
	double t_3 = t_2 * l;
	double tmp;
	if (t <= -1e-265) {
		tmp = 2.0 * ((t_3 / t_1) * (l / t_1));
	} else if (t <= -1e-300) {
		tmp = 2.0 * ((t_2 / t_1) * ((l * l) / t_1));
	} else if (t <= 0.0) {
		tmp = 2.0 * ((t_3 / (k * k)) * (l / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 2.0 * (Math.cos(k) * (1.0 / Math.pow((Math.sqrt(t) * (k * (Math.sin(k) / l))), 2.0)));
	}
	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 = k * math.sin(k)
	t_2 = math.cos(k) / t
	t_3 = t_2 * l
	tmp = 0
	if t <= -1e-265:
		tmp = 2.0 * ((t_3 / t_1) * (l / t_1))
	elif t <= -1e-300:
		tmp = 2.0 * ((t_2 / t_1) * ((l * l) / t_1))
	elif t <= 0.0:
		tmp = 2.0 * ((t_3 / (k * k)) * (l / math.pow(math.sin(k), 2.0)))
	else:
		tmp = 2.0 * (math.cos(k) * (1.0 / math.pow((math.sqrt(t) * (k * (math.sin(k) / l))), 2.0)))
	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(k * sin(k))
	t_2 = Float64(cos(k) / t)
	t_3 = Float64(t_2 * l)
	tmp = 0.0
	if (t <= -1e-265)
		tmp = Float64(2.0 * Float64(Float64(t_3 / t_1) * Float64(l / t_1)));
	elseif (t <= -1e-300)
		tmp = Float64(2.0 * Float64(Float64(t_2 / t_1) * Float64(Float64(l * l) / t_1)));
	elseif (t <= 0.0)
		tmp = Float64(2.0 * Float64(Float64(t_3 / Float64(k * k)) * Float64(l / (sin(k) ^ 2.0))));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(1.0 / (Float64(sqrt(t) * Float64(k * Float64(sin(k) / l))) ^ 2.0))));
	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 = k * sin(k);
	t_2 = cos(k) / t;
	t_3 = t_2 * l;
	tmp = 0.0;
	if (t <= -1e-265)
		tmp = 2.0 * ((t_3 / t_1) * (l / t_1));
	elseif (t <= -1e-300)
		tmp = 2.0 * ((t_2 / t_1) * ((l * l) / t_1));
	elseif (t <= 0.0)
		tmp = 2.0 * ((t_3 / (k * k)) * (l / (sin(k) ^ 2.0)));
	else
		tmp = 2.0 * (cos(k) * (1.0 / ((sqrt(t) * (k * (sin(k) / l))) ^ 2.0)));
	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[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * l), $MachinePrecision]}, If[LessEqual[t, -1e-265], N[(2.0 * N[(N[(t$95$3 / t$95$1), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1e-300], N[(2.0 * N[(N[(t$95$2 / t$95$1), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.0], N[(2.0 * N[(N[(t$95$3 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(1.0 / N[Power[N[(N[Sqrt[t], $MachinePrecision] * N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.99999999999999985e-266

   1. Initial program 47.3

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

   2. Simplified 40.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 8 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 0 points increase in error, 5 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.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))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in k around inf 22.4

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

   4. Applied egg-rr 23.8

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

   5. Applied egg-rr 23.7

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

   6. Applied egg-rr 6.4

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

   if -9.99999999999999985e-266 < t < -1.00000000000000003e-300

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 8 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 0 points increase in error, 5 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.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))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in k around inf 30.2

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

   4. Applied egg-rr 32.9

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

   5. Applied egg-rr 32.9

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

   6. Applied egg-rr 21.5

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

   if -1.00000000000000003e-300 < t < 0.0

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 8 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 0 points increase in error, 5 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.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))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in k around inf 31.0

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

   4. Applied egg-rr 35.7

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

   5. Applied egg-rr 35.7

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

   6. Applied egg-rr 30.2

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

   if 0.0 < t

   1. Initial program 47.0

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

   2. Simplified 40.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 2 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 8 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 0 points increase in error, 5 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 1 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.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))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in k around inf 23.1

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

   4. Applied egg-rr 3.4

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

   5. Taylor expanded in k around inf 2.5

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

   6. Simplified 0.9

      \[\leadsto 2 \cdot \left(\cos k \cdot \frac{1}{{\color{blue}{\left(\sqrt{t} \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right)}}^{2}}\right) \]
      Proof
      (*.f64 (sqrt.f64 t) (*.f64 (/.f64 (sin.f64 k) l) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 t) (Rewrite<= associate-/r/_binary64 (/.f64 (sin.f64 k) (/.f64 l k)))): 22 points increase in error, 16 points decrease in error
      (*.f64 (sqrt.f64 t) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sin.f64 k) k) l))): 14 points increase in error, 19 points decrease in error
      (*.f64 (sqrt.f64 t) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 k (sin.f64 k))) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 k (sin.f64 k)) l) (sqrt.f64 t))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{t} \cdot \ell}{k \cdot \sin k} \cdot \frac{\ell}{k \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{t}}{k \cdot \sin k} \cdot \frac{\ell \cdot \ell}{k \cdot \sin k}\right)\\ \mathbf{elif}\;t \leq 0:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{t} \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{1}{{\left(\sqrt{t} \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	26892

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := \frac{\cos k}{t}\\ t_3 := t_2 \cdot \ell\\ \mathbf{if}\;t \leq -1 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t_1} \cdot \frac{\ell \cdot \ell}{t_1}\right)\\ \mathbf{elif}\;t \leq 0:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{1}{{\left(\sqrt{t} \cdot \frac{t_1}{\ell}\right)}^{2}}\right)\\ \end{array} \]

Alternative 2
Error	5.6
Cost	26764

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := \frac{\cos k}{t}\\ t_3 := t_2 \cdot \ell\\ \mathbf{if}\;t \leq -1 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t_1} \cdot \frac{\ell \cdot \ell}{t_1}\right)\\ \mathbf{elif}\;t \leq 0:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot {\left(\frac{t_1 \cdot \sqrt{t}}{\ell}\right)}^{-2}\right)\\ \end{array} \]

Alternative 3
Error	5.6
Cost	26764

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := \frac{\cos k}{t}\\ t_3 := t_2 \cdot \ell\\ \mathbf{if}\;t \leq -1 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t_1} \cdot \frac{\ell \cdot \ell}{t_1}\right)\\ \mathbf{elif}\;t \leq 0:\\ \;\;\;\;2 \cdot \left(\frac{t_3}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot {\left(\frac{\ell}{t_1 \cdot \sqrt{t}}\right)}^{2}\right)\\ \end{array} \]

Alternative 4
Error	4.7
Cost	20884

\[\begin{array}{l} t_1 := \frac{\cos k}{t}\\ t_2 := 2 \cdot \left(\frac{t_1 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ t_3 := 2 \cdot \frac{t_1}{{\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{elif}\;k \leq 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	7.7
Cost	20684

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := \frac{\cos k}{t}\\ t_3 := 2 \cdot \left(\frac{t_2 \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{-265}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-300}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t_1} \cdot \frac{\ell \cdot \ell}{t_1}\right)\\ \mathbf{elif}\;t \leq 10^{-206}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2}{{\left(\frac{t_1}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 6
Error	5.7
Cost	20552

\[\begin{array}{l} t_1 := k \cdot \sin k\\ t_2 := 2 \cdot \left(\frac{\frac{\cos k}{t} \cdot \ell}{t_1} \cdot \frac{\ell}{t_1}\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	6.8
Cost	20232

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\cos k}{t}}{{\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	16.6
Cost	14408

\[\begin{array}{l} t_1 := \frac{\cos k}{k \cdot k} \cdot \frac{2 \cdot \frac{\ell}{\frac{t}{\ell}}}{0.5 + \cos \left(k + k\right) \cdot -0.5}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-6}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	25.1
Cost	7876

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{2}{k \cdot k} + 0.6666666666666666\right)\right)}{k}}{k}\\ \end{array} \]

Alternative 10
Error	25.7
Cost	7752

\[\begin{array}{l} t_1 := \frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{if}\;k \leq 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{+56}:\\ \;\;\;\;\frac{\frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}} \cdot \frac{2}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	25.4
Cost	7620

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\ell}{t \cdot \frac{k}{\frac{\ell}{k}}}\right)\\ \end{array} \]

Alternative 12
Error	28.1
Cost	960

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

Alternative 13
Error	27.1
Cost	960

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

Alternative 14
Error	27.1
Cost	960

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

Alternative 15
Error	26.6
Cost	960

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

Reproduce

herbie shell --seed 2022308 
(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))))