Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 1.8

Time: 31.7s

Precision: binary64

Cost: 20552

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2} \cdot t}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k \cdot \frac{\ell}{k \cdot t}}{\sin k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (if (<= k -2.6e-82)
   (* 2.0 (/ (/ (* (/ l k) (cos k)) (* (pow (sin k) 2.0) t)) (/ k l)))
   (if (<= k 1.65e-85)
     (* 2.0 (* (/ l k) (/ (/ (* (cos k) (/ l (* k t))) (sin k)) (sin k))))
     (* 2.0 (* (/ l k) (/ 1.0 (* (* (sin k) (tan k)) (* t (/ 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 tmp;
	if (k <= -2.6e-82) {
		tmp = 2.0 * ((((l / k) * cos(k)) / (pow(sin(k), 2.0) * t)) / (k / l));
	} else if (k <= 1.65e-85) {
		tmp = 2.0 * ((l / k) * (((cos(k) * (l / (k * t))) / sin(k)) / sin(k)));
	} else {
		tmp = 2.0 * ((l / k) * (1.0 / ((sin(k) * tan(k)) * (t * (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) :: tmp
    if (k <= (-2.6d-82)) then
        tmp = 2.0d0 * ((((l / k) * cos(k)) / ((sin(k) ** 2.0d0) * t)) / (k / l))
    else if (k <= 1.65d-85) then
        tmp = 2.0d0 * ((l / k) * (((cos(k) * (l / (k * t))) / sin(k)) / sin(k)))
    else
        tmp = 2.0d0 * ((l / k) * (1.0d0 / ((sin(k) * tan(k)) * (t * (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 tmp;
	if (k <= -2.6e-82) {
		tmp = 2.0 * ((((l / k) * Math.cos(k)) / (Math.pow(Math.sin(k), 2.0) * t)) / (k / l));
	} else if (k <= 1.65e-85) {
		tmp = 2.0 * ((l / k) * (((Math.cos(k) * (l / (k * t))) / Math.sin(k)) / Math.sin(k)));
	} else {
		tmp = 2.0 * ((l / k) * (1.0 / ((Math.sin(k) * Math.tan(k)) * (t * (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):
	tmp = 0
	if k <= -2.6e-82:
		tmp = 2.0 * ((((l / k) * math.cos(k)) / (math.pow(math.sin(k), 2.0) * t)) / (k / l))
	elif k <= 1.65e-85:
		tmp = 2.0 * ((l / k) * (((math.cos(k) * (l / (k * t))) / math.sin(k)) / math.sin(k)))
	else:
		tmp = 2.0 * ((l / k) * (1.0 / ((math.sin(k) * math.tan(k)) * (t * (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)
	tmp = 0.0
	if (k <= -2.6e-82)
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * cos(k)) / Float64((sin(k) ^ 2.0) * t)) / Float64(k / l)));
	elseif (k <= 1.65e-85)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(Float64(cos(k) * Float64(l / Float64(k * t))) / sin(k)) / sin(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(1.0 / Float64(Float64(sin(k) * tan(k)) * Float64(t * 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)
	tmp = 0.0;
	if (k <= -2.6e-82)
		tmp = 2.0 * ((((l / k) * cos(k)) / ((sin(k) ^ 2.0) * t)) / (k / l));
	elseif (k <= 1.65e-85)
		tmp = 2.0 * ((l / k) * (((cos(k) * (l / (k * t))) / sin(k)) / sin(k)));
	else
		tmp = 2.0 * ((l / k) * (1.0 / ((sin(k) * tan(k)) * (t * (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_] := If[LessEqual[k, -2.6e-82], N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.65e-85], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.6e-82

   1. Initial program 46.1

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

   2. Simplified 37.6

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 34 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)))): 53 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, 0 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, 53 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 19.7

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

   4. Simplified 7.2

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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)))): 68 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))))): 17 points increase in error, 24 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 0.7

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

   if -2.6e-82 < k < 1.64999999999999986e-85

   1. Initial program 63.9

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

   2. Simplified 61.9

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      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))))): 34 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)))): 53 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, 0 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, 53 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 53.7

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

   4. Simplified 32.0

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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)))): 68 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))))): 17 points increase in error, 24 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 29.0

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

   6. Applied egg-rr 28.2

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

   7. Applied egg-rr 12.7

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

   if 1.64999999999999986e-85 < k

   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 38.3

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (*.f64 (tan.f64 k) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1))))): 34 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)))): 53 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, 0 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, 53 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 20.0

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

   4. Simplified 7.3

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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)))): 68 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))))): 17 points increase in error, 24 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 1.3

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

   6. Applied egg-rr 0.6

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

   7. Applied egg-rr 0.6

      \[\leadsto 2 \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{1}{\left(\frac{\sin k}{1} \cdot \tan k\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2} \cdot t}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k \cdot \frac{\ell}{k \cdot t}}{\sin k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	20484

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{t \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \left(t \cdot \left(-\tan k\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	5.8
Cost	14156

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 0.000122:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	2.2
Cost	14152

\[\begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-155}:\\ \;\;\;\;-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \left(t \cdot \left(-\tan k\right)\right)}\right)\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\right)\\ \end{array} \]

Alternative 4
Error	2.2
Cost	14088

\[\begin{array}{l} t_1 := -2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \left(t \cdot \left(-\tan k\right)\right)}\right)\\ \mathbf{if}\;k \leq 2.45 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 0.000195:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	7.1
Cost	14024

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(\tan k \cdot \left(\sin k \cdot t\right)\right)}\right)\\ \mathbf{if}\;k \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 0.00033:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.1
Cost	14024

\[\begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \left(t \cdot \left(-\tan k\right)\right)\right)}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \end{array} \]

Alternative 7
Error	3.5
Cost	14024

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t \cdot \left(\sin k \cdot \tan k\right)}\right)\\ \mathbf{if}\;k \leq -9.8 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	25.0
Cost	8712

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-305}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{-123}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\left(t_1 \cdot -0.5 + \frac{\ell}{t \cdot {k}^{3}}\right) + t_1 \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 9
Error	25.0
Cost	8712

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-305}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{-123}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(t_1 \cdot -0.5 + \frac{\ell}{t \cdot {k}^{3}}\right) + t_1 \cdot 0.3333333333333333}{\frac{k}{\ell}}\\ \end{array} \]

Alternative 10
Error	24.9
Cost	1736

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{-123}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	26.3
Cost	1088

\[2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right) \]

Alternative 12
Error	32.7
Cost	704

\[-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right) \]

Alternative 13
Error	32.6
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell}{\frac{k}{\ell} \cdot \left(k \cdot t\right)} \]

Reproduce

herbie shell --seed 2022329 
(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))))