Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 7.2

Time: 48.4s

Precision: binary64

Cost: 52680

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{t_1 \cdot t}\\ \mathbf{if}\;k \leq -7 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 47000000:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{2 \cdot \ell}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+180}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{t_1}\right)\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0))
        (t_2 (* 2.0 (/ (* (/ (* (cos k) l) k) (/ l k)) (* t_1 t)))))
   (if (<= k -7e+48)
     t_2
     (if (<= k 47000000.0)
       (pow
        (/
         (*
          (/ (cbrt l) (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
          (cbrt (* 2.0 l)))
         (* t (cbrt (sin k))))
        3.0)
       (if (<= k 1.32e+124)
         (* (/ 2.0 (/ (* t (* k k)) l)) (/ (/ l (sin k)) (tan k)))
         (if (<= k 1.6e+180)
           t_2
           (* 2.0 (* (/ (/ (cos k) k) t) (/ (/ l (/ k l)) t_1)))))))))

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 = pow(sin(k), 2.0);
	double t_2 = 2.0 * ((((cos(k) * l) / k) * (l / k)) / (t_1 * t));
	double tmp;
	if (k <= -7e+48) {
		tmp = t_2;
	} else if (k <= 47000000.0) {
		tmp = pow((((cbrt(l) / cbrt((tan(k) * (2.0 + pow((k / t), 2.0))))) * cbrt((2.0 * l))) / (t * cbrt(sin(k)))), 3.0);
	} else if (k <= 1.32e+124) {
		tmp = (2.0 / ((t * (k * k)) / l)) * ((l / sin(k)) / tan(k));
	} else if (k <= 1.6e+180) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (((cos(k) / k) / t) * ((l / (k / l)) / t_1));
	}
	return tmp;
}

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.pow(Math.sin(k), 2.0);
	double t_2 = 2.0 * ((((Math.cos(k) * l) / k) * (l / k)) / (t_1 * t));
	double tmp;
	if (k <= -7e+48) {
		tmp = t_2;
	} else if (k <= 47000000.0) {
		tmp = Math.pow((((Math.cbrt(l) / Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))))) * Math.cbrt((2.0 * l))) / (t * Math.cbrt(Math.sin(k)))), 3.0);
	} else if (k <= 1.32e+124) {
		tmp = (2.0 / ((t * (k * k)) / l)) * ((l / Math.sin(k)) / Math.tan(k));
	} else if (k <= 1.6e+180) {
		tmp = t_2;
	} else {
		tmp = 2.0 * (((Math.cos(k) / k) / t) * ((l / (k / l)) / t_1));
	}
	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 = sin(k) ^ 2.0
	t_2 = Float64(2.0 * Float64(Float64(Float64(Float64(cos(k) * l) / k) * Float64(l / k)) / Float64(t_1 * t)))
	tmp = 0.0
	if (k <= -7e+48)
		tmp = t_2;
	elseif (k <= 47000000.0)
		tmp = Float64(Float64(Float64(cbrt(l) / cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) * cbrt(Float64(2.0 * l))) / Float64(t * cbrt(sin(k)))) ^ 3.0;
	elseif (k <= 1.32e+124)
		tmp = Float64(Float64(2.0 / Float64(Float64(t * Float64(k * k)) / l)) * Float64(Float64(l / sin(k)) / tan(k)));
	elseif (k <= 1.6e+180)
		tmp = t_2;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / k) / t) * Float64(Float64(l / Float64(k / l)) / t_1)));
	end
	return 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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -7e+48], t$95$2, If[LessEqual[k, 47000000.0], N[Power[N[(N[(N[(N[Power[l, 1/3], $MachinePrecision] / N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k, 1.32e+124], N[(N[(2.0 / N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+180], t$95$2, N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / N[(k / l), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -6.9999999999999995e48 or 1.32000000000000001e124 < k < 1.59999999999999997e180

   1. Initial program 32.5

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

   2. Simplified 32.5

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

   3. Taylor expanded in t around 0 20.1

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

   4. Simplified 21.5

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot t}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.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 (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.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 (*.f64 (cos.f64 k) (pow.f64 l 2)) (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
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 23 points increase in error, 11 points decrease in error

   5. Applied egg-rr 8.5

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

   if -6.9999999999999995e48 < k < 4.7e7

   1. Initial program 32.7

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

   2. Simplified 29.1

      \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{\frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\frac{\ell}{\tan k}}}} \]
      Proof
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 l (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (tan.f64 k)) l)))): 4 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (/.f64 l (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 (*.f64 (pow.f64 t 3) (sin.f64 k))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 20 points increase in error, 3 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (/.f64 (*.f64 l l) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))))): 4 points increase in error, 10 points decrease in error
      (/.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (*.f64 l l)))): 9 points increase in error, 12 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (*.f64 l l)) (*.f64 (tan.f64 k) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 7 points increase in error, 9 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l/_binary64 (*.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)))): 5 points increase in error, 3 points decrease in error
      (/.f64 2 (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)))): 1 points increase in error, 2 points decrease in error

   3. Applied egg-rr 21.5

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

   4. Applied egg-rr 12.2

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

   5. Simplified 12.1

      \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\ell \cdot 2}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
      Proof
      (*.f64 (cbrt.f64 (/.f64 l (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (cbrt.f64 (*.f64 l 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2))))) (cbrt.f64 (*.f64 l 2))): 2 points increase in error, 9 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (cbrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))) (Rewrite<= unpow1/3_binary64 (pow.f64 (*.f64 2 l) 1/3))): 167 points increase in error, 33 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 2 l) 1/3) (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/3_binary64 (cbrt.f64 (*.f64 2 l))) (cbrt.f64 (/.f64 (/.f64 l (tan.f64 k)) (+.f64 2 (pow.f64 (/.f64 k t) 2))))): 33 points increase in error, 167 points decrease in error

   6. Applied egg-rr 5.3

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

   7. Simplified 5.3

      \[\leadsto {\left(\frac{\color{blue}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3} \]
      Proof
      (/.f64 (cbrt.f64 l) (cbrt.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (cbrt.f64 l) 1)) (cbrt.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (cbrt.f64 l) (/.f64 1 (cbrt.f64 (*.f64 (tan.f64 k) (+.f64 2 (pow.f64 (/.f64 k t) 2))))))): 21 points increase in error, 22 points decrease in error

   if 4.7e7 < k < 1.32000000000000001e124

   1. Initial program 29.7

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

   2. Simplified 24.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}} \]
      Proof
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 2 (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2))))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) (/.f64 (/.f64 l (sin.f64 k)) (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (Rewrite<= associate-/r*_binary64 (/.f64 l (*.f64 (sin.f64 k) (tan.f64 k))))): 15 points increase in error, 1 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (/.f64 (pow.f64 t 3) l)) l) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k))))): 15 points increase in error, 13 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (/.f64 (pow.f64 t 3) l) l))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 6 points increase in error, 4 points decrease in error
      (/.f64 (/.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (sin.f64 k) (tan.f64 k)))): 31 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 2 (/.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))): 0 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.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, 4 points decrease in error
      (/.f64 (/.f64 2 (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)): 2 points increase in error, 35 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.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)))): 4 points increase in error, 3 points decrease in error

   3. Taylor expanded in t around 0 11.6

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

   4. Simplified 11.7

      \[\leadsto \color{blue}{\frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell}}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k} \]
      Proof
      (/.f64 2 (/.f64 (*.f64 (*.f64 k k) t) l)): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 l) (*.f64 (pow.f64 k 2) t))): 18 points increase in error, 19 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 l (*.f64 (pow.f64 k 2) t)))): 2 points increase in error, 1 points decrease in error

   if 1.59999999999999997e180 < k

   1. Initial program 32.9

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

   2. Simplified 32.9

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

   3. Taylor expanded in t around 0 23.0

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

   4. Simplified 23.0

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{{\sin k}^{2} \cdot t}} \]
      Proof
      (*.f64 2 (/.f64 (/.f64 (*.f64 (cos.f64 k) (*.f64 l l)) (*.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 (*.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.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 (*.f64 (cos.f64 k) (pow.f64 l 2)) (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
      (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 23 points increase in error, 11 points decrease in error

   5. Applied egg-rr 6.5

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -7 \cdot 10^{+48}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t}\\ \mathbf{elif}\;k \leq 47000000:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{2 \cdot \ell}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+180}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	46408

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{t_1 \cdot t}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10500000:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot \ell} \cdot \frac{1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k}{\ell}}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{elif}\;k \leq 3.45 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+180}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{t_1}\right)\\ \end{array} \]

Alternative 2
Error	8.4
Cost	46148

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\left(\frac{\frac{\sqrt[3]{\frac{2 \cdot \ell}{\sin k}}}{t}}{\sqrt[3]{\tan k \cdot \frac{t_1}{\ell}}}\right)}^{3}\\ \mathbf{if}\;t \leq -2.65 \cdot 10^{-24}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot \ell} \cdot \sqrt[3]{\frac{\ell}{\tan k \cdot t_1}}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{elif}\;t \leq 105000:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+286}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k \cdot t}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	8.4
Cost	40144

\[\begin{array}{l} t_1 := {\left(\frac{\sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{\frac{t}{\sqrt[3]{2 \cdot \frac{\ell}{\sin k}}}}\right)}^{3}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 72000:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+287}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k \cdot t}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.4
Cost	40144

\[\begin{array}{l} t_1 := {\left(\frac{\frac{\sqrt[3]{\frac{2 \cdot \ell}{\sin k}}}{t}}{\sqrt[3]{\tan k \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\ell}}}\right)}^{3}\\ \mathbf{if}\;t \leq -1.46 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 25000:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+286}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k \cdot t}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	9.1
Cost	33744

\[\begin{array}{l} t_1 := \frac{2 \cdot \ell}{\sin k}\\ t_2 := \frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ t_3 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{{t_3}^{2}}{t}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{t_1}{t \cdot t}}{t} \cdot t_2\\ \mathbf{elif}\;t \leq 30500:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+154}:\\ \;\;\;\;t_2 \cdot {\left(\frac{\sqrt[3]{t_1}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t_3}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 6
Error	9.0
Cost	21136

\[\begin{array}{l} t_1 := \frac{\frac{\frac{2 \cdot \ell}{\sin k}}{t \cdot t}}{t} \cdot \frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+119}:\\ \;\;\;\;\frac{{t_2}^{2}}{t}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 25000:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t_2}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 7
Error	11.3
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{t_1 \cdot t}\\ \mathbf{if}\;k \leq -0.004:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-22}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k \cdot t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+125}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{elif}\;k \leq 10^{+180}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\frac{\ell}{\frac{k}{\ell}}}{t_1}\right)\\ \end{array} \]

Alternative 8
Error	11.1
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\cos k \cdot \ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t}\\ \mathbf{if}\;k \leq -11000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-22}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k \cdot t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	15.2
Cost	14024

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -3700000:\\ \;\;\;\;\frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot k\right)}{\ell}} \cdot \frac{\frac{\ell}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t_1}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 10
Error	19.7
Cost	13960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t_1}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 11
Error	19.4
Cost	13576

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-32}:\\ \;\;\;\;\frac{{t_1}^{2}}{t}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t_1}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 12
Error	20.0
Cost	7304

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{k \cdot t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{t_1}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 13
Error	29.1
Cost	832

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

Alternative 14
Error	28.9
Cost	832

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

Alternative 15
Error	23.7
Cost	832

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t}\\ t_1 \cdot \frac{t_1}{t} \end{array} \]

Alternative 16
Error	23.7
Cost	832

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

Reproduce

herbie shell --seed 2022318 
(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))))