# ?

Average Error: 0.1 → 0.1
Time: 12.8s
Precision: binary64
Cost: 7424

# ?

$0 \leq x \land x \leq 10^{+18}$
$x \cdot \left(\frac{15}{8} - \left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)$
$x \cdot 1.875 - \left(x \cdot x\right) \cdot \left({x}^{3} \cdot -0.375 + x \cdot 1.25\right)$
(FPCore (x)
:precision binary64
(*
x
(-
(/ 15.0 8.0)
(*
(* (sqrt (/ 3.0 8.0)) (pow x 2.0))
(- (sqrt (/ 25.0 6.0)) (* (sqrt (/ 3.0 8.0)) (pow x 2.0)))))))
(FPCore (x)
:precision binary64
(- (* x 1.875) (* (* x x) (+ (* (pow x 3.0) -0.375) (* x 1.25)))))
double code(double x) {
return x * ((15.0 / 8.0) - ((sqrt((3.0 / 8.0)) * pow(x, 2.0)) * (sqrt((25.0 / 6.0)) - (sqrt((3.0 / 8.0)) * pow(x, 2.0)))));
}

double code(double x) {
return (x * 1.875) - ((x * x) * ((pow(x, 3.0) * -0.375) + (x * 1.25)));
}

real(8) function code(x)
real(8), intent (in) :: x
code = x * ((15.0d0 / 8.0d0) - ((sqrt((3.0d0 / 8.0d0)) * (x ** 2.0d0)) * (sqrt((25.0d0 / 6.0d0)) - (sqrt((3.0d0 / 8.0d0)) * (x ** 2.0d0)))))
end function

real(8) function code(x)
real(8), intent (in) :: x
code = (x * 1.875d0) - ((x * x) * (((x ** 3.0d0) * (-0.375d0)) + (x * 1.25d0)))
end function

public static double code(double x) {
return x * ((15.0 / 8.0) - ((Math.sqrt((3.0 / 8.0)) * Math.pow(x, 2.0)) * (Math.sqrt((25.0 / 6.0)) - (Math.sqrt((3.0 / 8.0)) * Math.pow(x, 2.0)))));
}

public static double code(double x) {
return (x * 1.875) - ((x * x) * ((Math.pow(x, 3.0) * -0.375) + (x * 1.25)));
}

def code(x):
return x * ((15.0 / 8.0) - ((math.sqrt((3.0 / 8.0)) * math.pow(x, 2.0)) * (math.sqrt((25.0 / 6.0)) - (math.sqrt((3.0 / 8.0)) * math.pow(x, 2.0)))))

def code(x):
return (x * 1.875) - ((x * x) * ((math.pow(x, 3.0) * -0.375) + (x * 1.25)))

function code(x)
return Float64(x * Float64(Float64(15.0 / 8.0) - Float64(Float64(sqrt(Float64(3.0 / 8.0)) * (x ^ 2.0)) * Float64(sqrt(Float64(25.0 / 6.0)) - Float64(sqrt(Float64(3.0 / 8.0)) * (x ^ 2.0))))))
end

function code(x)
return Float64(Float64(x * 1.875) - Float64(Float64(x * x) * Float64(Float64((x ^ 3.0) * -0.375) + Float64(x * 1.25))))
end

function tmp = code(x)
tmp = x * ((15.0 / 8.0) - ((sqrt((3.0 / 8.0)) * (x ^ 2.0)) * (sqrt((25.0 / 6.0)) - (sqrt((3.0 / 8.0)) * (x ^ 2.0)))));
end

function tmp = code(x)
tmp = (x * 1.875) - ((x * x) * (((x ^ 3.0) * -0.375) + (x * 1.25)));
end

code[x_] := N[(x * N[(N[(15.0 / 8.0), $MachinePrecision] - N[(N[(N[Sqrt[N[(3.0 / 8.0),$MachinePrecision]], $MachinePrecision] * N[Power[x, 2.0],$MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(25.0 / 6.0),$MachinePrecision]], $MachinePrecision] - N[(N[Sqrt[N[(3.0 / 8.0),$MachinePrecision]], $MachinePrecision] * N[Power[x, 2.0],$MachinePrecision]), $MachinePrecision]),$MachinePrecision]), $MachinePrecision]),$MachinePrecision]), $MachinePrecision]  code[x_] := N[(N[(x * 1.875),$MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(N[(N[Power[x, 3.0],$MachinePrecision] * -0.375), $MachinePrecision] + N[(x * 1.25),$MachinePrecision]), $MachinePrecision]),$MachinePrecision]), \$MachinePrecision]

x \cdot \left(\frac{15}{8} - \left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)

x \cdot 1.875 - \left(x \cdot x\right) \cdot \left({x}^{3} \cdot -0.375 + x \cdot 1.25\right)


# Try it out?

Your Program's Arguments

Results

 In Out
Enter valid numbers for all inputs

# Derivation?

1. Initial program 0.1

$x \cdot \left(\frac{15}{8} - \left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)$
2. Simplified0.1

$\leadsto \color{blue}{x \cdot 1.875 - \mathsf{fma}\left(\sqrt{0.375}, \sqrt{4.166666666666667}, \left(x \cdot x\right) \cdot -0.375\right) \cdot {x}^{3}}$
Proof
[Start]0.1 $x \cdot \left(\frac{15}{8} - \left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)$ $\color{blue}{x \cdot \frac{15}{8} - x \cdot \left(\left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)}$ $x \cdot \color{blue}{1.875} - x \cdot \left(\left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)$ $x \cdot 1.875 - \color{blue}{\left(\left(\sqrt{\frac{3}{8}} \cdot {x}^{2}\right) \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right) \cdot x}$ $x \cdot 1.875 - \left(\color{blue}{\left({x}^{2} \cdot \sqrt{\frac{3}{8}}\right)} \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right) \cdot x$ $x \cdot 1.875 - \color{blue}{\left({x}^{2} \cdot \left(\sqrt{\frac{3}{8}} \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right)\right)} \cdot x$ $x \cdot 1.875 - \color{blue}{\left(\left(\sqrt{\frac{3}{8}} \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)} \cdot x$ $x \cdot 1.875 - \color{blue}{\left(\sqrt{\frac{3}{8}} \cdot \left(\sqrt{\frac{25}{6}} - \sqrt{\frac{3}{8}} \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} \cdot x\right)}$
3. Applied egg-rr0.1

$\leadsto x \cdot 1.875 - \color{blue}{\left({x}^{3} \cdot \left(\left(x \cdot x\right) \cdot -0.375\right) + {x}^{3} \cdot 1.25\right)}$
4. Applied egg-rr0.1

$\leadsto x \cdot 1.875 - \color{blue}{\left(x \cdot x\right) \cdot \left({x}^{3} \cdot -0.375 + 1.25 \cdot x\right)}$
5. Final simplification0.1

$\leadsto x \cdot 1.875 - \left(x \cdot x\right) \cdot \left({x}^{3} \cdot -0.375 + x \cdot 1.25\right)$

# Alternatives

Alternative 1
Error0.1
Cost7360
$x \cdot 1.875 - \left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.375, 1.25\right)\right)$
Alternative 2
Error0.1
Cost7296
$x \cdot 1.875 + {x}^{3} \cdot \left(-1.25 + \left(x \cdot x\right) \cdot 0.375\right)$
Alternative 3
Error0.1
Cost7296
$x \cdot 1.875 + {x}^{3} \cdot \left(-1.25 + x \cdot \left(x \cdot 0.375\right)\right)$
Alternative 4
Error0.1
Cost960
$x \cdot \left(1.875 + \left(x \cdot x\right) \cdot \left(-1.25 - \left(x \cdot x\right) \cdot -0.375\right)\right)$
Alternative 5
Error3.9
Cost192
$x \cdot 1.875$

# Reproduce?

herbie shell --seed 1
(FPCore (x)
:name "x*(15/8-((sqrt(3/8)*x^2)*(sqrt(25/6)-(sqrt(3/8)*x^2))))"
:precision binary64
:pre (and (<= 0.0 x) (<= x 1e+18))
(* x (- (/ 15.0 8.0) (* (* (sqrt (/ 3.0 8.0)) (pow x 2.0)) (- (sqrt (/ 25.0 6.0)) (* (sqrt (/ 3.0 8.0)) (pow x 2.0)))))))