?

Average Error: 3.9 → 3.4
Time: 10.8s
Precision: binary64
Cost: 45376

?

\[-10 \leq x \land x \leq 10\]
\[{x}^{\left(\frac{1}{7}\right)} \]
↓
\[{\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \left(\sqrt[3]{{x}^{0.011904761904761904}} \cdot \sqrt[3]{{x}^{0.005952380952380952}}\right)\right)}^{8} \]
(FPCore (x) :precision binary64 (pow x (/ 1.0 7.0)))
↓
(FPCore (x)
 :precision binary64
 (pow
  (*
   (cbrt (pow x 0.03571428571428571))
   (* (cbrt (pow x 0.011904761904761904)) (cbrt (pow x 0.005952380952380952))))
  8.0))
double code(double x) {
	return pow(x, (1.0 / 7.0));
}
↓
double code(double x) {
	return pow((cbrt(pow(x, 0.03571428571428571)) * (cbrt(pow(x, 0.011904761904761904)) * cbrt(pow(x, 0.005952380952380952)))), 8.0);
}

public static double code(double x) {
	return Math.pow(x, (1.0 / 7.0));
}
↓
public static double code(double x) {
	return Math.pow((Math.cbrt(Math.pow(x, 0.03571428571428571)) * (Math.cbrt(Math.pow(x, 0.011904761904761904)) * Math.cbrt(Math.pow(x, 0.005952380952380952)))), 8.0);
}

function code(x)
	return x ^ Float64(1.0 / 7.0)
end
↓
function code(x)
	return Float64(cbrt((x ^ 0.03571428571428571)) * Float64(cbrt((x ^ 0.011904761904761904)) * cbrt((x ^ 0.005952380952380952)))) ^ 8.0
end

code[x_] := N[Power[x, N[(1.0 / 7.0), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Power[N[(N[Power[N[Power[x, 0.03571428571428571], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[Power[x, 0.011904761904761904], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[x, 0.005952380952380952], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 8.0], $MachinePrecision]

{x}^{\left(\frac{1}{7}\right)}
↓
{\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \left(\sqrt[3]{{x}^{0.011904761904761904}} \cdot \sqrt[3]{{x}^{0.005952380952380952}}\right)\right)}^{8}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 3.9

    \[{x}^{\left(\frac{1}{7}\right)} \]

  2. Simplified3.9

    \[\leadsto \color{blue}{{\left({x}^{0.017857142857142856}\right)}^{8}} \]
    Proof

    [Start]3.9

    \[ {x}^{\left(\frac{1}{7}\right)} \]

    sqr-pow [=>]3.9

    \[ \color{blue}{{x}^{\left(\frac{\frac{1}{7}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{7}}{2}\right)}} \]

    sqr-pow [=>]3.9

    \[ {x}^{\left(\frac{\frac{1}{7}}{2}\right)} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)}\right)} \]

    associate-*r* [=>]3.9

    \[ \color{blue}{\left({x}^{\left(\frac{\frac{1}{7}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)}} \]

    *-commutative [<=]3.9

    \[ \color{blue}{\left({x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{7}}{2}\right)}\right)} \cdot {x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)} \]

    sqr-pow [=>]3.9

    \[ \left({x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{7}}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{\frac{\frac{\frac{1}{7}}{2}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{\frac{\frac{1}{7}}{2}}{2}}{2}\right)}\right)} \]

    associate-*r* [=>]3.9

    \[ \color{blue}{\left(\left({x}^{\left(\frac{\frac{\frac{1}{7}}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{7}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{\frac{\frac{1}{7}}{2}}{2}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{\frac{\frac{1}{7}}{2}}{2}}{2}\right)}} \]

  3. Applied egg-rr3.5

    \[\leadsto {\color{blue}{\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \sqrt[3]{{x}^{0.017857142857142856}}\right)}}^{8} \]

  4. Applied egg-rr3.4

    \[\leadsto {\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \color{blue}{\left(\sqrt[3]{{x}^{0.011904761904761904}} \cdot \sqrt[3]{{x}^{0.005952380952380952}}\right)}\right)}^{8} \]

  5. Final simplification3.4

    \[\leadsto {\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \left(\sqrt[3]{{x}^{0.011904761904761904}} \cdot \sqrt[3]{{x}^{0.005952380952380952}}\right)\right)}^{8} \]

Alternatives

Alternative 1
Error3.5
Cost45376
\[\begin{array}{l} t_0 := \sqrt[3]{{x}^{0.008928571428571428}}\\ {\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \left(t_0 \cdot t_0\right)\right)}^{8} \end{array} \]
Alternative 2
Error3.4
Cost45248
\[{\left(\sqrt[3]{\sqrt[3]{{\left({x}^{0.03571428571428571}\right)}^{3}}} \cdot \sqrt[3]{{x}^{0.017857142857142856}}\right)}^{8} \]
Alternative 3
Error3.5
Cost38848
\[{\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \sqrt[3]{e^{0.017857142857142856 \cdot \log x}}\right)}^{8} \]
Alternative 4
Error3.5
Cost38784
\[\sqrt{{\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \sqrt[3]{{x}^{0.017857142857142856}}\right)}^{16}} \]
Alternative 5
Error3.5
Cost32384
\[{\left(\sqrt[3]{{x}^{0.03571428571428571}} \cdot \sqrt[3]{{x}^{0.017857142857142856}}\right)}^{8} \]
Alternative 6
Error3.6
Cost25856
\[{\left({\left(\sqrt[3]{{x}^{0.03571428571428571}}\right)}^{3}\right)}^{4} \]
Alternative 7
Error3.6
Cost19456
\[e^{4 \cdot \log \left({x}^{0.03571428571428571}\right)} \]
Alternative 8
Error3.8
Cost19392
\[\sqrt{e^{\log x \cdot 0.2857142857142857}} \]
Alternative 9
Error3.8
Cost12992
\[e^{\log x \cdot 0.14285714285714285} \]
Alternative 10
Error3.9
Cost6528
\[{x}^{0.14285714285714285} \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "x^(1/7)"
  :precision binary64
  :pre (and (<= -10.0 x) (<= x 10.0))
  (pow x (/ 1.0 7.0)))