Average Error: 47.5 → 1.6
Time: 36.2s
Precision: binary64
Cost: 20488
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2}}}{t}}{\frac{k}{\ell}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-110}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(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
         (* 2.0 (/ (/ (/ (* (/ l k) (cos k)) (pow (sin k) 2.0)) t) (/ k l)))))
   (if (<= k -1e-145)
     t_1
     (if (<= k 1e-110)
       (* 2.0 (* (* (/ l k) (/ l k)) (/ 1.0 (* k (* k t)))))
       t_1))))
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 = 2.0 * (((((l / k) * cos(k)) / pow(sin(k), 2.0)) / t) / (k / l));
	double tmp;
	if (k <= -1e-145) {
		tmp = t_1;
	} else if (k <= 1e-110) {
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = 2.0d0 * (((((l / k) * cos(k)) / (sin(k) ** 2.0d0)) / t) / (k / l))
    if (k <= (-1d-145)) then
        tmp = t_1
    else if (k <= 1d-110) then
        tmp = 2.0d0 * (((l / k) * (l / k)) * (1.0d0 / (k * (k * t))))
    else
        tmp = t_1
    end if
    code = tmp
end function

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 = 2.0 * (((((l / k) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0)) / t) / (k / l));
	double tmp;
	if (k <= -1e-145) {
		tmp = t_1;
	} else if (k <= 1e-110) {
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = 2.0 * (((((l / k) * math.cos(k)) / math.pow(math.sin(k), 2.0)) / t) / (k / l))
	tmp = 0
	if k <= -1e-145:
		tmp = t_1
	elif k <= 1e-110:
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))))
	else:
		tmp = t_1
	return tmp

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(2.0 * Float64(Float64(Float64(Float64(Float64(l / k) * cos(k)) / (sin(k) ^ 2.0)) / t) / Float64(k / l)))
	tmp = 0.0
	if (k <= -1e-145)
		tmp = t_1;
	elseif (k <= 1e-110)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(1.0 / Float64(k * Float64(k * t)))));
	else
		tmp = t_1;
	end
	return tmp
end

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 = 2.0 * (((((l / k) * cos(k)) / (sin(k) ^ 2.0)) / t) / (k / l));
	tmp = 0.0;
	if (k <= -1e-145)
		tmp = t_1;
	elseif (k <= 1e-110)
		tmp = 2.0 * (((l / k) * (l / k)) * (1.0 / (k * (k * t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

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[(2.0 * N[(N[(N[(N[(N[(l / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1e-145], t$95$1, If[LessEqual[k, 1e-110], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$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)}
↓
\begin{array}{l}
t_1 := 2 \cdot \frac{\frac{\frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2}}}{t}}{\frac{k}{\ell}}\\
\mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;k \leq 10^{-110}:\\
\;\;\;\;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}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < -9.99999999999999915e-146 or 1.0000000000000001e-110 < k

    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. Simplified38.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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))): 4 points increase in error, 2 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))): 5 points increase in error, 5 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, 4 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, 4 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)))): 31 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)))): 7 points increase in error, 11 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))): 6 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))))): 6 points increase in error, 7 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)))): 4 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))): 0 points increase in error, 2 points decrease in error

    3. Applied egg-rr34.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\left(\tan k \cdot 0.5\right) \cdot \frac{\sin k}{{t}^{-3}}}}{\frac{k}{t}}}{\frac{k}{t}}} \]

    4. Taylor expanded in l around 0 21.0

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

    5. Simplified7.8

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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, 15 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))))): 30 points increase in error, 15 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

    6. Applied egg-rr1.7

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

    7. Applied egg-rr0.9

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

    if -9.99999999999999915e-146 < k < 1.0000000000000001e-110

    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. Simplified63.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      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))): 4 points increase in error, 2 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))): 5 points increase in error, 5 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, 4 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, 4 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)))): 31 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)))): 7 points increase in error, 11 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))): 6 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))))): 6 points increase in error, 7 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)))): 4 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))): 0 points increase in error, 2 points decrease in error

    3. Applied egg-rr61.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{\left(\tan k \cdot 0.5\right) \cdot \frac{\sin k}{{t}^{-3}}}}{\frac{k}{t}}}{\frac{k}{t}}} \]

    4. Taylor expanded in l around 0 59.2

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

    5. Simplified39.8

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      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, 15 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))))): 30 points increase in error, 15 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

    6. Taylor expanded in k around 0 39.8

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

    7. Simplified16.4

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{1}{k \cdot \left(k \cdot t\right)}}\right) \]
      Proof
      (/.f64 1 (*.f64 k (*.f64 k t))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t))): 35 points increase in error, 32 points decrease in error
      (/.f64 1 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t)): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2}}}{t}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 10^{-110}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\frac{\ell}{k} \cdot \cos k}{{\sin k}^{2}}}{t}}{\frac{k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error1.8
Cost20488
\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\frac{\cos k}{{\sin k}^{2}}}{t \cdot \frac{k}{\ell}}}{\frac{k}{\ell}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-90}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error1.7
Cost20488
\[\begin{array}{l} t_1 := \frac{\cos k}{{\sin k}^{2}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-145}:\\ \;\;\;\;2 \cdot \frac{\frac{t_1}{t \cdot \frac{k}{\ell}}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 10^{-76}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(t_1 \cdot \frac{\frac{\ell}{k}}{t}\right)\right)\\ \end{array} \]
Alternative 3
Error3.5
Cost20484
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+172}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\cos k}{{\sin k}^{2}}}{k \cdot t}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}}{\frac{k}{\ell}}\\ \end{array} \]
Alternative 4
Error3.0
Cost14408
\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}}{\frac{k}{\ell}}\\ \mathbf{if}\;k \leq -1000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-10}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(k \cdot t\right)}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error23.6
Cost1608
\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{k \cdot \left(k \cdot t\right)} + \frac{-0.16666666666666666}{t}\right)\right)\\ \mathbf{if}\;t \leq -0.0014405552053337203:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.833105885307894 \cdot 10^{-34}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error25.1
Cost1220
\[\begin{array}{l} \mathbf{if}\;t \leq -8.455594133649624 \cdot 10^{+168}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]
Alternative 7
Error27.4
Cost960
\[\frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)} \]
Alternative 8
Error26.4
Cost960
\[\frac{\frac{2}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}}{k} \]
Alternative 9
Error26.4
Cost960
\[\frac{\ell \cdot \frac{2}{k}}{k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)} \]
Alternative 10
Error26.1
Cost960
\[\frac{\ell \cdot \frac{\frac{2}{k}}{k}}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} \]

Error

Reproduce

herbie shell --seed 2022316 
(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))))