Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 1.4

Time: 45.6s

Precision: binary64

Cost: 33220

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -4 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k}{t_1}}\right)}^{3}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot t_1} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t}\right)\\ \end{array} \]

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)))
   (if (<= k -4e-144)
     (* 2.0 (/ (pow (cbrt (* (/ (/ l k) t) (/ (cos k) t_1))) 3.0) (/ k l)))
     (if (<= k 4.9e-99)
       (* 2.0 (/ (/ (* (cos k) (pow (/ l k) 2.0)) (* k t)) k))
       (* 2.0 (* (/ l (* k t_1)) (/ (* (/ l k) (cos k)) t)))))))

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 tmp;
	if (k <= -4e-144) {
		tmp = 2.0 * (pow(cbrt((((l / k) / t) * (cos(k) / t_1))), 3.0) / (k / l));
	} else if (k <= 4.9e-99) {
		tmp = 2.0 * (((cos(k) * pow((l / k), 2.0)) / (k * t)) / k);
	} else {
		tmp = 2.0 * ((l / (k * t_1)) * (((l / k) * cos(k)) / t));
	}
	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 tmp;
	if (k <= -4e-144) {
		tmp = 2.0 * (Math.pow(Math.cbrt((((l / k) / t) * (Math.cos(k) / t_1))), 3.0) / (k / l));
	} else if (k <= 4.9e-99) {
		tmp = 2.0 * (((Math.cos(k) * Math.pow((l / k), 2.0)) / (k * t)) / k);
	} else {
		tmp = 2.0 * ((l / (k * t_1)) * (((l / k) * Math.cos(k)) / t));
	}
	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
	tmp = 0.0
	if (k <= -4e-144)
		tmp = Float64(2.0 * Float64((cbrt(Float64(Float64(Float64(l / k) / t) * Float64(cos(k) / t_1))) ^ 3.0) / Float64(k / l)));
	elseif (k <= 4.9e-99)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * (Float64(l / k) ^ 2.0)) / Float64(k * t)) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * t_1)) * Float64(Float64(Float64(l / k) * cos(k)) / t)));
	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]}, If[LessEqual[k, -4e-144], N[(2.0 * N[(N[Power[N[Power[N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.9e-99], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -3.9999999999999998e-144

   1. Initial program 46.6

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

   2. Simplified 38.4

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 39 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)))): 37 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))): 2 points increase in error, 3 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, 37 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 21.1

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

   4. Simplified 7.0

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 63 points increase in error, 20 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 17 points increase in error, 13 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (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

   5. Applied egg-rr 1.6

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

   6. Applied egg-rr 1.3

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

   if -3.9999999999999998e-144 < k < 4.9000000000000003e-99

   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 62.8

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 39 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)))): 37 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))): 2 points increase in error, 3 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, 37 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 58.8

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

   4. Simplified 39.5

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 63 points increase in error, 20 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 17 points increase in error, 13 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (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

   5. Taylor expanded in k around 0 39.5

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

   6. Simplified 18.0

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]
      Proof
      (*.f64 k (*.f64 k t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t)): 57 points increase in error, 18 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 9.0

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

   if 4.9000000000000003e-99 < k

   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 38.7

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 39 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)))): 37 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))): 2 points increase in error, 3 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, 37 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 19.2

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

   4. Simplified 7.6

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 63 points increase in error, 20 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 l 2) (cos.f64 k)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 17 points increase in error, 13 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (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

   5. Applied egg-rr 10.3

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

   6. Applied egg-rr 0.6

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \frac{{\left(\sqrt[3]{\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}\right)}^{3}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot {\sin k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	20488

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k \cdot {\sin k}^{2}} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t}\right)\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	5.1
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}}{\frac{k}{\ell}}\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-246}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}}{t_2}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-199}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 3.9 \cdot 10^{+116}:\\ \;\;\;\;2 \cdot \frac{1}{t_2 \cdot \left(\left(k \cdot t\right) \cdot \frac{\frac{k}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.8
Cost	14284

\[\begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := 2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}}{t_1}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{-247}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-200}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(\left(k \cdot t\right) \cdot \frac{\frac{k}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	8.8
Cost	14024

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}}{\sin k \cdot \tan k}\\ \mathbf{if}\;\ell \leq -8.4 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-202}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.1
Cost	7620

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-196}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\frac{1}{t}}{k \cdot k} + \frac{-0.5}{t}\right)\right)\\ \end{array} \]

Alternative 6
Error	23.8
Cost	7620

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-198}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{k}}{k \cdot t}\right)}{k}\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\frac{1}{t}}{k \cdot k} + \frac{-0.5}{t}\right)\right)\\ \end{array} \]

Alternative 7
Error	23.7
Cost	7620

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-200}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{\frac{k}{\ell}}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\frac{1}{t}}{k \cdot k} + \frac{-0.5}{t}\right)\right)\\ \end{array} \]

Alternative 8
Error	24.1
Cost	7432

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-247}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{k \cdot k}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\frac{1}{t}}{k \cdot k} + \frac{-0.5}{t}\right)\right)\\ \end{array} \]

Alternative 9
Error	25.0
Cost	1224

\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{k \cdot k}\\ \mathbf{if}\;k \leq -2.2 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell}}}{k} \cdot \frac{\frac{2}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	25.0
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -1.12 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell}}}{k} \cdot \frac{\frac{2}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{k \cdot k}\\ \end{array} \]

Alternative 11
Error	25.0
Cost	1220

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+139}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k}}{t \cdot \frac{k}{\ell}}}{k \cdot k}\\ \end{array} \]

Alternative 12
Error	27.0
Cost	960

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

Alternative 13
Error	25.7
Cost	960

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

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))))