Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 7.5

Time: 39.2s

Precision: binary64

Cost: 27020

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\frac{\ell}{k}\right)}^{2}\\ t_3 := \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{t_1}\right) \cdot \left(2 \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{if}\;k \leq -3.624322661808349 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)}{t}}{t_1}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 10^{-52}:\\ \;\;\;\;\frac{t_2 \cdot \left(\cos k \cdot 2\right)}{\sin k} \cdot \frac{\frac{1}{t}}{\sin k}\\ \mathbf{elif}\;k \leq 1.079557095758532 \cdot 10^{+195}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot t_2\right)}{t_1}}{t}\\ \end{array} \]

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 (pow (/ l k) 2.0))
        (t_3 (/ (* (* (/ l t) (/ l t_1)) (* 2.0 (/ (cos k) k))) k)))
   (if (<= k -3.624322661808349e+162)
     (/ (/ (* (cos k) (* 2.0 (/ (/ l k) (/ k l)))) t) t_1)
     (if (<= k -1e-150)
       t_3
       (if (<= k 1e-52)
         (* (/ (* t_2 (* (cos k) 2.0)) (sin k)) (/ (/ 1.0 t) (sin k)))
         (if (<= k 1.079557095758532e+195)
           t_3
           (/ (/ (* (cos k) (* 2.0 t_2)) t_1) 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 t_2 = pow((l / k), 2.0);
	double t_3 = (((l / t) * (l / t_1)) * (2.0 * (cos(k) / k))) / k;
	double tmp;
	if (k <= -3.624322661808349e+162) {
		tmp = ((cos(k) * (2.0 * ((l / k) / (k / l)))) / t) / t_1;
	} else if (k <= -1e-150) {
		tmp = t_3;
	} else if (k <= 1e-52) {
		tmp = ((t_2 * (cos(k) * 2.0)) / sin(k)) * ((1.0 / t) / sin(k));
	} else if (k <= 1.079557095758532e+195) {
		tmp = t_3;
	} else {
		tmp = ((cos(k) * (2.0 * t_2)) / t_1) / t;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = (l / k) ** 2.0d0
    t_3 = (((l / t) * (l / t_1)) * (2.0d0 * (cos(k) / k))) / k
    if (k <= (-3.624322661808349d+162)) then
        tmp = ((cos(k) * (2.0d0 * ((l / k) / (k / l)))) / t) / t_1
    else if (k <= (-1d-150)) then
        tmp = t_3
    else if (k <= 1d-52) then
        tmp = ((t_2 * (cos(k) * 2.0d0)) / sin(k)) * ((1.0d0 / t) / sin(k))
    else if (k <= 1.079557095758532d+195) then
        tmp = t_3
    else
        tmp = ((cos(k) * (2.0d0 * t_2)) / t_1) / t
    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 t_2 = Math.pow((l / k), 2.0);
	double t_3 = (((l / t) * (l / t_1)) * (2.0 * (Math.cos(k) / k))) / k;
	double tmp;
	if (k <= -3.624322661808349e+162) {
		tmp = ((Math.cos(k) * (2.0 * ((l / k) / (k / l)))) / t) / t_1;
	} else if (k <= -1e-150) {
		tmp = t_3;
	} else if (k <= 1e-52) {
		tmp = ((t_2 * (Math.cos(k) * 2.0)) / Math.sin(k)) * ((1.0 / t) / Math.sin(k));
	} else if (k <= 1.079557095758532e+195) {
		tmp = t_3;
	} else {
		tmp = ((Math.cos(k) * (2.0 * t_2)) / t_1) / t;
	}
	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)
	t_2 = math.pow((l / k), 2.0)
	t_3 = (((l / t) * (l / t_1)) * (2.0 * (math.cos(k) / k))) / k
	tmp = 0
	if k <= -3.624322661808349e+162:
		tmp = ((math.cos(k) * (2.0 * ((l / k) / (k / l)))) / t) / t_1
	elif k <= -1e-150:
		tmp = t_3
	elif k <= 1e-52:
		tmp = ((t_2 * (math.cos(k) * 2.0)) / math.sin(k)) * ((1.0 / t) / math.sin(k))
	elif k <= 1.079557095758532e+195:
		tmp = t_3
	else:
		tmp = ((math.cos(k) * (2.0 * t_2)) / t_1) / 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
	t_2 = Float64(l / k) ^ 2.0
	t_3 = Float64(Float64(Float64(Float64(l / t) * Float64(l / t_1)) * Float64(2.0 * Float64(cos(k) / k))) / k)
	tmp = 0.0
	if (k <= -3.624322661808349e+162)
		tmp = Float64(Float64(Float64(cos(k) * Float64(2.0 * Float64(Float64(l / k) / Float64(k / l)))) / t) / t_1);
	elseif (k <= -1e-150)
		tmp = t_3;
	elseif (k <= 1e-52)
		tmp = Float64(Float64(Float64(t_2 * Float64(cos(k) * 2.0)) / sin(k)) * Float64(Float64(1.0 / t) / sin(k)));
	elseif (k <= 1.079557095758532e+195)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(cos(k) * Float64(2.0 * t_2)) / t_1) / t);
	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;
	t_2 = (l / k) ^ 2.0;
	t_3 = (((l / t) * (l / t_1)) * (2.0 * (cos(k) / k))) / k;
	tmp = 0.0;
	if (k <= -3.624322661808349e+162)
		tmp = ((cos(k) * (2.0 * ((l / k) / (k / l)))) / t) / t_1;
	elseif (k <= -1e-150)
		tmp = t_3;
	elseif (k <= 1e-52)
		tmp = ((t_2 * (cos(k) * 2.0)) / sin(k)) * ((1.0 / t) / sin(k));
	elseif (k <= 1.079557095758532e+195)
		tmp = t_3;
	else
		tmp = ((cos(k) * (2.0 * t_2)) / t_1) / t;
	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]}, Block[{t$95$2 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(l / t), $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, -3.624322661808349e+162], N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[k, -1e-150], t$95$3, If[LessEqual[k, 1e-52], N[(N[(N[(t$95$2 * N[(N[Cos[k], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / t), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.079557095758532e+195], t$95$3, N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -3.62432266180834935e162

   1. Initial program 38.2

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

   2. Applied egg-rr 33.0

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

   3. Taylor expanded in t around 0 24.1

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

   4. Simplified 4.1

      \[\leadsto \color{blue}{\frac{\cos k}{{\sin k}^{2} \cdot t} \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot 2\right)} \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) 2)): 70 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (/.f64 (pow.f64 l 2) (pow.f64 k 2))) 2)): 1 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) (pow.f64 k 2)))) 2): 20 points increase in error, 17 points decrease in error
      (*.f64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.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 4.1

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

   6. Applied egg-rr 4.1

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

   if -3.62432266180834935e162 < k < -1.00000000000000001e-150 or 1e-52 < k < 1.07955709575853201e195

   1. Initial program 51.3

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

   2. Simplified 41.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof
      (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 4 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 34 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 5 points increase in error, 1 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 0 points increase in error, 5 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 1 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 2 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 0 points increase in error, 1 points decrease in error

   3. Taylor expanded in k around inf 18.4

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

   4. Simplified 18.4

      \[\leadsto \color{blue}{\frac{\cos k}{k \cdot k} \cdot \left(\frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}} \cdot 2\right)} \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 k k)) (*.f64 (/.f64 (/.f64 (*.f64 l l) t) (pow.f64 (sin.f64 k) 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) (*.f64 (/.f64 (/.f64 (*.f64 l l) t) (pow.f64 (sin.f64 k) 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t) (pow.f64 (sin.f64 k) 2)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) 2)): 10 points increase in error, 11 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (cos.f64 k) (pow.f64 k 2)) (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) 2)): 2 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 22 points increase in error, 23 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.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 38.6

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

   6. Taylor expanded in k around inf 16.4

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

   7. Simplified 9.2

      \[\leadsto \frac{\color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \left(\frac{\cos k}{k} \cdot 2\right)}}{k} \]
      Proof
      (*.f64 (*.f64 (/.f64 l t) (/.f64 l (pow.f64 (sin.f64 k) 2))) (*.f64 (/.f64 (cos.f64 k) k) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t (pow.f64 (sin.f64 k) 2)))) (*.f64 (/.f64 (cos.f64 k) k) 2)): 46 points increase in error, 29 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t (pow.f64 (sin.f64 k) 2))) (*.f64 (/.f64 (cos.f64 k) k) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (/.f64 (cos.f64 k) k) 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (/.f64 (cos.f64 k) k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cos.f64 k) k) (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 k (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 22 points increase in error, 32 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 k (*.f64 (pow.f64 (sin.f64 k) 2) t))))): 0 points increase in error, 0 points decrease in error

   if -1.00000000000000001e-150 < k < 1e-52

   1. Initial program 63.4

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

   2. Applied egg-rr 59.7

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

   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 32.1

      \[\leadsto \color{blue}{\frac{\cos k}{{\sin k}^{2} \cdot t} \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot 2\right)} \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) 2)): 70 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (/.f64 (pow.f64 l 2) (pow.f64 k 2))) 2)): 1 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) (pow.f64 k 2)))) 2): 20 points increase in error, 17 points decrease in error
      (*.f64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.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 31.3

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

   6. Applied egg-rr 12.0

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

   if 1.07955709575853201e195 < k

   1. Initial program 38.9

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

   2. Applied egg-rr 34.3

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

   3. Taylor expanded in t around 0 23.6

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

   4. Simplified 3.4

      \[\leadsto \color{blue}{\frac{\cos k}{{\sin k}^{2} \cdot t} \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot 2\right)} \]
      Proof
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) 2)): 70 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k)) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (*.f64 (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2))) 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (cos.f64 k) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (/.f64 (pow.f64 l 2) (pow.f64 k 2))) 2)): 1 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (*.f64 (*.f64 (pow.f64 (sin.f64 k) 2) t) (pow.f64 k 2)))) 2): 20 points increase in error, 17 points decrease in error
      (*.f64 (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)))) 2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.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 3.3

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

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.624322661808349 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)}{t}}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \left(2 \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{elif}\;k \leq 10^{-52}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\cos k \cdot 2\right)}{\sin k} \cdot \frac{\frac{1}{t}}{\sin k}\\ \mathbf{elif}\;k \leq 1.079557095758532 \cdot 10^{+195}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \left(2 \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)}{{\sin k}^{2}}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	26960

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{t_1}\right) \cdot \left(2 \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{if}\;k \leq -3.624322661808349 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)}{t}}{t_1}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-98}:\\ \;\;\;\;\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;k \leq 1.702805684086891 \cdot 10^{+187}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;{\sin k}^{-2} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\cos k \cdot 2\right)}{t}\\ \end{array} \]

Alternative 2
Error	7.8
Cost	26960

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{t_1}\right) \cdot \left(2 \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{if}\;k \leq -3.624322661808349 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)}{t}}{t_1}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-98}:\\ \;\;\;\;\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;k \leq 1.079557095758532 \cdot 10^{+195}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)}{t_1}}{t}\\ \end{array} \]

Alternative 3
Error	8.3
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{t_1}\right) \cdot \left(2 \cdot \frac{\cos k}{k}\right)}{k}\\ \mathbf{if}\;k \leq -3.624322661808349 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)}{t}}{t_1}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-98}:\\ \;\;\;\;\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;k \leq 1.079557095758532 \cdot 10^{+195}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\right)}{t}}{t_1}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)}{t}}{t_1}\\ \mathbf{elif}\;k \leq 1.079557095758532 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{\cos k}{k} \cdot \left({\sin k}^{-2} \cdot \left(\ell \cdot \left(2 \cdot \frac{\ell}{t}\right)\right)\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot \frac{\ell \cdot \frac{\ell}{k}}{k}\right)}{t}}{t_1}\\ \end{array} \]

Alternative 5
Error	9.1
Cost	20224

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

Alternative 6
Error	7.0
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ t_2 := t_1 \cdot \frac{\cos k}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ t_3 := \frac{2}{k \cdot k}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\ell \cdot t_3}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 10^{-90}:\\ \;\;\;\;\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot t_1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{\ell}{t \cdot \left(k \cdot \frac{k}{\ell}\right)} \cdot \left(t_3 + 0.6666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	23.9
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot \left(\frac{k}{\ell} \cdot t\right)} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-90}:\\ \;\;\;\;\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	28.9
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{k}}{k}}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot -0.3333333333333333}{\frac{k}{\ell} \cdot t}\\ \end{array} \]

Alternative 9
Error	26.1
Cost	1220

\[\begin{array}{l} t_1 := \frac{2}{k \cdot k}\\ \mathbf{if}\;\ell \leq -1.714697811744225 \cdot 10^{-201}:\\ \;\;\;\;\left(t_1 + -0.3333333333333333\right) \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot t_1}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 10
Error	30.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot -0.3333333333333333}{\frac{k}{\ell} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\frac{t}{\ell}}\\ \end{array} \]

Alternative 11
Error	25.9
Cost	1088

\[\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{k}}{k}\right) \cdot \left(\frac{2}{k \cdot k} + 0.3333333333333333\right) \]

Alternative 12
Error	25.2
Cost	1088

\[\frac{\ell}{k \cdot \left(\frac{k}{\ell} \cdot t\right)} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right) \]

Alternative 13
Error	26.5
Cost	960

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

Alternative 14
Error	32.5
Cost	704

\[\frac{\frac{\ell}{k} \cdot -0.3333333333333333}{\frac{k}{\ell} \cdot t} \]

Reproduce

herbie shell --seed 2022317 
(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))))