Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 1.8

Time: 29.0s

Precision: binary64

Cost: 14152

Math

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

↓

\[\begin{array}{l} t_1 := \sin k \cdot t\\ t_2 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{\tan k}}{t_1}\right)\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{t_1}\right) \cdot \frac{1}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (* (sin k) t))
        (t_2 (* (/ l k) (* (/ l k) (/ (/ 2.0 (tan k)) t_1)))))
   (if (<= k -3.5e-102)
     t_2
     (if (<= k 8e-142)
       (* (* (* (/ l k) (/ l k)) (/ 2.0 t_1)) (/ 1.0 (tan k)))
       t_2))))

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 = sin(k) * t;
	double t_2 = (l / k) * ((l / k) * ((2.0 / tan(k)) / t_1));
	double tmp;
	if (k <= -3.5e-102) {
		tmp = t_2;
	} else if (k <= 8e-142) {
		tmp = (((l / k) * (l / k)) * (2.0 / t_1)) * (1.0 / tan(k));
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = sin(k) * t
    t_2 = (l / k) * ((l / k) * ((2.0d0 / tan(k)) / t_1))
    if (k <= (-3.5d-102)) then
        tmp = t_2
    else if (k <= 8d-142) then
        tmp = (((l / k) * (l / k)) * (2.0d0 / t_1)) * (1.0d0 / tan(k))
    else
        tmp = t_2
    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.sin(k) * t;
	double t_2 = (l / k) * ((l / k) * ((2.0 / Math.tan(k)) / t_1));
	double tmp;
	if (k <= -3.5e-102) {
		tmp = t_2;
	} else if (k <= 8e-142) {
		tmp = (((l / k) * (l / k)) * (2.0 / t_1)) * (1.0 / Math.tan(k));
	} else {
		tmp = t_2;
	}
	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.sin(k) * t
	t_2 = (l / k) * ((l / k) * ((2.0 / math.tan(k)) / t_1))
	tmp = 0
	if k <= -3.5e-102:
		tmp = t_2
	elif k <= 8e-142:
		tmp = (((l / k) * (l / k)) * (2.0 / t_1)) * (1.0 / math.tan(k))
	else:
		tmp = t_2
	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 = Float64(sin(k) * t)
	t_2 = Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(2.0 / tan(k)) / t_1)))
	tmp = 0.0
	if (k <= -3.5e-102)
		tmp = t_2;
	elseif (k <= 8e-142)
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 / t_1)) * Float64(1.0 / tan(k)));
	else
		tmp = t_2;
	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) * t;
	t_2 = (l / k) * ((l / k) * ((2.0 / tan(k)) / t_1));
	tmp = 0.0;
	if (k <= -3.5e-102)
		tmp = t_2;
	elseif (k <= 8e-142)
		tmp = (((l / k) * (l / k)) * (2.0 / t_1)) * (1.0 / tan(k));
	else
		tmp = t_2;
	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[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -3.5e-102], t$95$2, If[LessEqual[k, 8e-142], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if k < -3.49999999999999986e-102 or 8.0000000000000003e-142 < k

   1. Initial program 46.8

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

   2. Simplified 39.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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, 0 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, 3 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))): 0 points increase in error, 0 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))): 2 points increase in error, 1 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)))): 38 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)))): 2 points increase in error, 2 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, 1 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))))): 2 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)))): 0 points increase in error, 3 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))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 35.4

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

   4. Taylor expanded in t around 0 14.9

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

   5. Simplified 11.9

      \[\leadsto \frac{\frac{2}{\tan k} \cdot \ell}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \left(\sin k \cdot t\right)}} \]
      Proof
      (*.f64 (/.f64 (*.f64 k k) l) (*.f64 (sin.f64 k) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) l) (*.f64 (sin.f64 k) t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 k 2) (/.f64 l (*.f64 (sin.f64 k) t)))): 20 points increase in error, 28 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (sin.f64 k) t)) l)): 31 points increase in error, 27 points decrease in error

   6. Applied egg-rr 5.9

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

   7. Applied egg-rr 1.5

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

   if -3.49999999999999986e-102 < k < 8.0000000000000003e-142

   1. Initial program 64.0

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

   2. Simplified 63.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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, 0 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, 3 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))): 0 points increase in error, 0 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))): 2 points increase in error, 1 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)))): 38 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)))): 2 points increase in error, 2 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, 1 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))))): 2 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)))): 0 points increase in error, 3 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))): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 59.4

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

   4. Taylor expanded in t around 0 48.4

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

   5. Simplified 39.0

      \[\leadsto \frac{\frac{2}{\tan k} \cdot \ell}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \left(\sin k \cdot t\right)}} \]
      Proof
      (*.f64 (/.f64 (*.f64 k k) l) (*.f64 (sin.f64 k) t)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) l) (*.f64 (sin.f64 k) t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 k 2) (/.f64 l (*.f64 (sin.f64 k) t)))): 20 points increase in error, 28 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (sin.f64 k) t)) l)): 31 points increase in error, 27 points decrease in error

   6. Applied egg-rr 41.0

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

   7. Applied egg-rr 9.4

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

   8. Simplified 6.2

      \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot t}\right) \cdot \frac{1}{\tan k}} \]
      Proof
      (*.f64 (*.f64 (*.f64 (/.f64 l k) (/.f64 l k)) (/.f64 2 (*.f64 (sin.f64 k) t))) (/.f64 1 (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k))) (/.f64 2 (*.f64 (sin.f64 k) t))) (/.f64 1 (tan.f64 k))): 62 points increase in error, 7 points decrease in error
      (*.f64 (*.f64 (Rewrite=> associate-/l*_binary64 (/.f64 l (/.f64 (*.f64 k k) l))) (/.f64 2 (*.f64 (sin.f64 k) t))) (/.f64 1 (tan.f64 k))): 5 points increase in error, 46 points decrease in error
      (*.f64 (*.f64 (/.f64 l (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 k l) k))) (/.f64 2 (*.f64 (sin.f64 k) t))) (/.f64 1 (tan.f64 k))): 5 points increase in error, 31 points decrease in error
      (*.f64 (*.f64 (/.f64 l (Rewrite<= associate-/r/_binary64 (/.f64 k (/.f64 l k)))) (/.f64 2 (*.f64 (sin.f64 k) t))) (/.f64 1 (tan.f64 k))): 7 points increase in error, 8 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l 2) (*.f64 (/.f64 k (/.f64 l k)) (*.f64 (sin.f64 k) t)))) (/.f64 1 (tan.f64 k))): 19 points increase in error, 37 points decrease in error
      (*.f64 (/.f64 (*.f64 l 2) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sin.f64 k) t) (/.f64 k (/.f64 l k))))) (/.f64 1 (tan.f64 k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 l (*.f64 (*.f64 (sin.f64 k) t) (/.f64 k (/.f64 l k)))) 2)) (/.f64 1 (tan.f64 k))): 2 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 (tan.f64 k)) (*.f64 (/.f64 l (*.f64 (*.f64 (sin.f64 k) t) (/.f64 k (/.f64 l k)))) 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 (tan.f64 k) (*.f64 (/.f64 l (*.f64 (*.f64 (sin.f64 k) t) (/.f64 k (/.f64 l k)))) 2)))): 20 points increase in error, 12 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot t}\right)\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot t}\right) \cdot \frac{1}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.9
Cost	14024

\[\begin{array}{l} t_1 := \ell \cdot \left(\frac{2}{\tan k} \cdot \frac{\frac{\ell}{k}}{k \cdot \left(\sin k \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -2 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\frac{\ell}{k \cdot k}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	2.8
Cost	14024

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{\tan k}}{\sin k \cdot t}\right)\\ \mathbf{if}\;k \leq -1.86 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\frac{\ell}{k \cdot k}}}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	23.6
Cost	7752

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k} \cdot 2}{\left(\sin k \cdot t\right) \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{if}\;k \leq -1.05 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{\frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	23.3
Cost	7752

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := \frac{2}{\frac{k \cdot \frac{t}{\frac{\ell}{k \cdot k}}}{\ell \cdot t_1}}\\ \mathbf{if}\;k \leq -3.4 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	24.4
Cost	7432

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell \cdot 2}{k \cdot k}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k}}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 6
Error	24.9
Cost	7360

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

Alternative 7
Error	26.3
Cost	960

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

Alternative 8
Error	25.8
Cost	960

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

Alternative 9
Error	33.0
Cost	832

\[\frac{0}{k} \cdot \frac{\frac{\ell}{k \cdot t}}{k \cdot k} \]

Alternative 10
Error	35.1
Cost	704

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

Reproduce

herbie shell --seed 2022337 
(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))))