?

Average Error: 47.9 → 24.4
Time: 35.8s
Precision: binary64
Cost: 32896

?

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)} \]
(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
 (* 2.0 (/ (* (pow l 2.0) (cos k)) (* t (* (pow k 2.0) (pow (sin k) 2.0))))))
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) {
	return 2.0 * ((pow(l, 2.0) * cos(k)) / (t * (pow(k, 2.0) * pow(sin(k), 2.0))));
}

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
    code = 2.0d0 * (((l ** 2.0d0) * cos(k)) / (t * ((k ** 2.0d0) * (sin(k) ** 2.0d0))))
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) {
	return 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (t * (Math.pow(k, 2.0) * Math.pow(Math.sin(k), 2.0))));
}

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):
	return 2.0 * ((math.pow(l, 2.0) * math.cos(k)) / (t * (math.pow(k, 2.0) * math.pow(math.sin(k), 2.0))))

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)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64(t * Float64((k ^ 2.0) * (sin(k) ^ 2.0)))))
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 = code(t, l, k)
	tmp = 2.0 * (((l ^ 2.0) * cos(k)) / (t * ((k ^ 2.0) * (sin(k) ^ 2.0))));
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_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 47.9

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

  2. Simplified40.5

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

    [Start]47.9

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

    rational_best_45_simplify-91 [=>]47.9

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

    rational_best_45_simplify-25 [=>]47.9

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

    rational_best_45_simplify-109 [=>]40.5

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

    metadata-eval [=>]40.5

    \[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right) \cdot \tan k\right)} \]

    rational_best_45_simplify-3 [=>]40.5

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

  3. Taylor expanded in t around 0 23.4

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

  4. Applied egg-rr23.4

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} + 0\right)} \]

  5. Simplified24.4

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

    [Start]23.4

    \[ 2 \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} + 0\right) \]

    rational_best_45_simplify-3 [=>]23.4

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

    rational_best_45_simplify-91 [=>]23.4

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

    rational_best_45_simplify-91 [=>]23.4

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

    rational_best_45_simplify-25 [=>]24.4

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

  6. Final simplification24.4

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

Alternatives

Alternative 1
Error29.2
Cost53700
\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\\ \mathbf{if}\;t_2 \cdot \left(\left(1 + t_1\right) - 1\right) \leq \infty:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t_1 + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot -0.16666666666666666\right)\\ \end{array} \]
Alternative 2
Error23.4
Cost32896
\[2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
Alternative 3
Error30.7
Cost26952
\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+89}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}\\ \end{array} \]
Alternative 4
Error30.7
Cost26952
\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}\\ \end{array} \]
Alternative 5
Error30.9
Cost26952
\[\begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+68}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}\\ \end{array} \]
Alternative 6
Error31.8
Cost26816
\[2 \cdot \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} + \frac{{\ell}^{2}}{t \cdot {k}^{2}} \cdot -0.16666666666666666\right) \]
Alternative 7
Error31.9
Cost26752
\[2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + \left(-\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right) \]
Alternative 8
Error31.2
Cost26496
\[2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({k}^{2} \cdot t\right)} \]
Alternative 9
Error32.3
Cost19904
\[2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{4} \cdot t} \]
Alternative 10
Error32.3
Cost19904
\[2 \cdot \frac{{\ell}^{2}}{t \cdot \left({k}^{3} \cdot \sin k\right)} \]
Alternative 11
Error31.6
Cost19904
\[2 \cdot \frac{{\ell}^{2}}{{k}^{3} \cdot \left(\sin k \cdot t\right)} \]
Alternative 12
Error32.4
Cost13376
\[2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]

Error

Reproduce?

herbie shell --seed 2023098 
(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))))