Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 1.8

Time: 39.2s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.4 \cdot 10^{-103}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{1}{t}}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\cos k}{t_1}}{t \cdot \frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \]

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 -1.4e-103)
     (* 2.0 (* (/ l k) (/ (* (/ l k) (cos k)) (* t t_1))))
     (if (<= k 2.2e-56)
       (* 2.0 (/ (* (pow (/ l k) 2.0) (/ (/ 1.0 t) (sin k))) (tan k)))
       (* 2.0 (/ (/ (/ (cos k) t_1) (* t (/ k l))) (/ k l)))))))

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 <= -1.4e-103) {
		tmp = 2.0 * ((l / k) * (((l / k) * cos(k)) / (t * t_1)));
	} else if (k <= 2.2e-56) {
		tmp = 2.0 * ((pow((l / k), 2.0) * ((1.0 / t) / sin(k))) / tan(k));
	} else {
		tmp = 2.0 * (((cos(k) / t_1) / (t * (k / l))) / (k / l));
	}
	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) ** 2.0d0
    if (k <= (-1.4d-103)) then
        tmp = 2.0d0 * ((l / k) * (((l / k) * cos(k)) / (t * t_1)))
    else if (k <= 2.2d-56) then
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) * ((1.0d0 / t) / sin(k))) / tan(k))
    else
        tmp = 2.0d0 * (((cos(k) / t_1) / (t * (k / l))) / (k / l))
    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.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= -1.4e-103) {
		tmp = 2.0 * ((l / k) * (((l / k) * Math.cos(k)) / (t * t_1)));
	} else if (k <= 2.2e-56) {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) * ((1.0 / t) / Math.sin(k))) / Math.tan(k));
	} else {
		tmp = 2.0 * (((Math.cos(k) / t_1) / (t * (k / l))) / (k / l));
	}
	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.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= -1.4e-103:
		tmp = 2.0 * ((l / k) * (((l / k) * math.cos(k)) / (t * t_1)))
	elif k <= 2.2e-56:
		tmp = 2.0 * ((math.pow((l / k), 2.0) * ((1.0 / t) / math.sin(k))) / math.tan(k))
	else:
		tmp = 2.0 * (((math.cos(k) / t_1) / (t * (k / l))) / (k / l))
	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 <= -1.4e-103)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(Float64(l / k) * cos(k)) / Float64(t * t_1))));
	elseif (k <= 2.2e-56)
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) * Float64(Float64(1.0 / t) / sin(k))) / tan(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / t_1) / Float64(t * Float64(k / l))) / Float64(k / l)));
	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) ^ 2.0;
	tmp = 0.0;
	if (k <= -1.4e-103)
		tmp = 2.0 * ((l / k) * (((l / k) * cos(k)) / (t * t_1)));
	elseif (k <= 2.2e-56)
		tmp = 2.0 * ((((l / k) ^ 2.0) * ((1.0 / t) / sin(k))) / tan(k));
	else
		tmp = 2.0 * (((cos(k) / t_1) / (t * (k / l))) / (k / l));
	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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, -1.4e-103], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e-56], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.40000000000000011e-103

   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 38.7

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 27 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)))): 39 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))): 1 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, 39 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 19.6

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

   4. Simplified 6.9

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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)))): 67 points increase in error, 8 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))))): 15 points increase in error, 14 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 1.1

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

   6. Applied egg-rr 0.6

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

   7. Applied egg-rr 39.6

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

   8. Simplified 1.0

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\cos k \cdot \frac{\ell}{k}}{t \cdot {\sin k}^{2}}\right)} \]
      Proof
      (*.f64 (/.f64 l k) (/.f64 (*.f64 (cos.f64 k) (/.f64 l k)) (*.f64 t (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 l k) (cos.f64 k))) (*.f64 t (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (cos.f64 k)) k)) (*.f64 t (pow.f64 (sin.f64 k) 2)))): 10 points increase in error, 13 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) l)) k) (*.f64 t (pow.f64 (sin.f64 k) 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (cos.f64 k) (/.f64 k l))) (*.f64 t (pow.f64 (sin.f64 k) 2)))): 12 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (cos.f64 k) (/.f64 k l)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite<= associate-/r*_binary64 (/.f64 (cos.f64 k) (*.f64 (/.f64 k l) (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 26 points increase in error, 19 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (cos.f64 k) (*.f64 (/.f64 k l) (Rewrite=> *-commutative_binary64 (*.f64 t (pow.f64 (sin.f64 k) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (cos.f64 k) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 k l) t) (pow.f64 (sin.f64 k) 2))))): 18 points increase in error, 32 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (cos.f64 k) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2)))): 15 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (Rewrite<= unpow1_binary64 (pow.f64 (cos.f64 k) 1)) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (cos.f64 k) (/.f64 1 2)) (pow.f64 (cos.f64 k) (/.f64 1 2)))) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2))): 104 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (*.f64 (pow.f64 (cos.f64 k) (Rewrite=> metadata-eval 1/2)) (pow.f64 (cos.f64 k) (/.f64 1 2))) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (cos.f64 k))) (pow.f64 (cos.f64 k) (/.f64 1 2))) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (*.f64 (sqrt.f64 (cos.f64 k)) (pow.f64 (cos.f64 k) (Rewrite=> metadata-eval 1/2))) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (/.f64 (*.f64 (sqrt.f64 (cos.f64 k)) (Rewrite=> unpow1/2_binary64 (sqrt.f64 (cos.f64 k)))) (*.f64 (/.f64 k l) t)) (pow.f64 (sin.f64 k) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (*.f64 (/.f64 k l) t)) (sqrt.f64 (cos.f64 k)))) (pow.f64 (sin.f64 k) 2))): 7 points increase in error, 6 points decrease in error
      (*.f64 (/.f64 l k) (/.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (*.f64 (/.f64 k l) t)) (sqrt.f64 (cos.f64 k))) (Rewrite=> unpow2_binary64 (*.f64 (sin.f64 k) (sin.f64 k))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (*.f64 (/.f64 k l) t)) (sin.f64 k)) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k))))): 18 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (sqrt.f64 (cos.f64 k)) (*.f64 (*.f64 (/.f64 k l) t) (sin.f64 k)))) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 18 points increase in error, 19 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (*.f64 (/.f64 k l) t))) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 17 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (/.f64 k l)) t)) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 13 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) k) l)) t) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 22 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) l) k)) t) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 12 points increase in error, 22 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (/.f64 l k))) t) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 15 points increase in error, 9 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 l k) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))) t) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 l k) t) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))): 13 points increase in error, 19 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite=> associate-*l*_binary64 (*.f64 (/.f64 (/.f64 l k) t) (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)))))): 27 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k))) (/.f64 (/.f64 l k) t)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l k) (*.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k))) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 (/.f64 (/.f64 l k) t)) (sqrt.f64 (/.f64 (/.f64 l k) t)))))): 67 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 l k) (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t))) (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t)))))): 5 points increase in error, 15 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 l k) (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t)))) (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t))))): 6 points increase in error, 10 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t))) (/.f64 l k))) (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t))) (*.f64 (*.f64 (/.f64 (sqrt.f64 (cos.f64 k)) (sin.f64 k)) (sqrt.f64 (/.f64 (/.f64 l k) t))) (/.f64 l k)))): 0 points increase in error, 0 points decrease in error

   if -1.40000000000000011e-103 < k < 2.20000000000000004e-56

   1. Initial program 63.6

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

   2. Simplified 61.2

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 27 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)))): 39 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))): 1 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, 39 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 48.4

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

   4. Simplified 30.2

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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)))): 67 points increase in error, 8 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))))): 15 points increase in error, 14 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 31.5

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

   6. Applied egg-rr 30.2

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

   7. Applied egg-rr 11.3

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

   if 2.20000000000000004e-56 < k

   1. Initial program 45.4

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

   2. Simplified 37.2

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 27 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)))): 39 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))): 1 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, 39 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 19.3

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

   4. Simplified 6.3

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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)))): 67 points increase in error, 8 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))))): 15 points increase in error, 14 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 0.8

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

   6. Applied egg-rr 0.4

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.4 \cdot 10^{-103}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-56}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{1}{t}}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\cos k}{{\sin k}^{2}}}{t \cdot \frac{k}{\ell}}}{\frac{k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	20488

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -4 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{1}{t}}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.8
Cost	20360

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{\ell}{k}}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k}{\ell}}\\ \mathbf{if}\;k \leq -1.95 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{1}{t}}{\sin k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	16.9
Cost	14288

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := 2 \cdot \left(\frac{\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right) \cdot -0.16666666666666666}{k \cdot k} + t_1 \cdot \frac{t_1}{t}\right)\\ t_3 := 2 \cdot \frac{\ell \cdot \ell}{\left(\sin k \cdot \left(t \cdot \tan k\right)\right) \cdot \left(k \cdot k\right)}\\ \mathbf{if}\;k \leq -0.0048:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -8 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	8.5
Cost	14288

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := 2 \cdot \left(\frac{\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right) \cdot -0.16666666666666666}{k \cdot k} + t_1 \cdot \frac{t_1}{t}\right)\\ t_3 := 2 \cdot \frac{\ell \cdot \frac{\ell}{k}}{k \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)}\\ \mathbf{if}\;k \leq -0.006:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -2.3 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	5.7
Cost	14288

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := 2 \cdot \frac{\ell \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \left(t \cdot \tan k\right)}}{k}\\ \mathbf{if}\;k \leq -3.9 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{k}\right)}^{2}}{t \cdot 0.5}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right) \cdot -0.16666666666666666}{k \cdot k} + t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	1.6
Cost	14220

\[\begin{array}{l} t_1 := t \cdot \tan k\\ t_2 := 2 \cdot \frac{\frac{\frac{\ell}{k}}{\sin k \cdot t_1}}{\frac{k}{\ell}}\\ \mathbf{if}\;k \leq -2.65 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{k}\right)}^{2}}{t \cdot 0.5}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{\sin k}}{t_1}}{k} \cdot \frac{\ell}{-k}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	2.4
Cost	14156

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{\ell}{k}}{\sin k \cdot \left(t \cdot \tan k\right)}}{\frac{k}{\ell}}\\ \mathbf{if}\;k \leq -2.5 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.4 \cdot 10^{-169}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{k}\right)}^{2}}{t \cdot 0.5}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	22.7
Cost	13704

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+47}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} + \frac{-0.16666666666666666}{t}\right)\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot 0.5}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}}{k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{\sqrt{\frac{2}{t}}}{k}\right)}^{2}\\ \end{array} \]

Alternative 9
Error	23.3
Cost	2120

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := 2 \cdot \left(\frac{\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right) \cdot -0.16666666666666666}{k \cdot k} + t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{if}\;k \leq -1.4 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-81}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	22.8
Cost	1608

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} + \frac{-0.16666666666666666}{t}\right)\right)\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot 0.5}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	23.9
Cost	1484

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{+227}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot 0.5}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}}{k}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+291}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \end{array} \]

Alternative 12
Error	23.8
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+89}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot 0.5}}{\frac{k}{\frac{\frac{\ell}{k}}{k}}}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\frac{1}{k}}{k \cdot t}\right)\\ \end{array} \]

Alternative 13
Error	24.0
Cost	1220

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot 0.5}}{k \cdot k}}{k \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 14
Error	24.1
Cost	1220

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-245}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot 0.5}}{k \cdot \frac{k}{\ell}}}{k \cdot k}\\ \end{array} \]

Alternative 15
Error	25.9
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \frac{2}{t} \cdot \left(t_1 \cdot t_1\right) \end{array} \]

Alternative 16
Error	25.0
Cost	960

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k}\\ \frac{2}{t} \cdot \left(t_1 \cdot t_1\right) \end{array} \]

Reproduce

herbie shell --seed 2022338 
(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))))