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(FPCore (y x) :precision binary64 (- y (/ (- (* y y) x) (* 2.0 y))))

(FPCore (y x) :precision binary64 (* 0.5 (+ y (/ x y))))

double code(double y, double x) {
	return y - (((y * y) - x) / (2.0 * y));
}

double code(double y, double x) {
	return 0.5 * (y + (x / y));
}

real(8) function code(y, x)
    real(8), intent (in) :: y
    real(8), intent (in) :: x
    code = y - (((y * y) - x) / (2.0d0 * y))
end function

real(8) function code(y, x)
    real(8), intent (in) :: y
    real(8), intent (in) :: x
    code = 0.5d0 * (y + (x / y))
end function

public static double code(double y, double x) {
	return y - (((y * y) - x) / (2.0 * y));
}

public static double code(double y, double x) {
	return 0.5 * (y + (x / y));
}

def code(y, x):
	return y - (((y * y) - x) / (2.0 * y))

def code(y, x):
	return 0.5 * (y + (x / y))

function code(y, x)
	return Float64(y - Float64(Float64(Float64(y * y) - x) / Float64(2.0 * y)))
end

function code(y, x)
	return Float64(0.5 * Float64(y + Float64(x / y)))
end

function tmp = code(y, x)
	tmp = y - (((y * y) - x) / (2.0 * y));
end

function tmp = code(y, x)
	tmp = 0.5 * (y + (x / y));
end

code[y_, x_] := N[(y - N[(N[(N[(y * y), $MachinePrecision] - x), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

code[y_, x_] := N[(0.5 * N[(y + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

y - \frac{y \cdot y - x}{2 \cdot y}

0.5 \cdot \left(y + \frac{x}{y}\right)

Error?

Try it out?

Derivation?

1. Initial program 18.0

   \[y - \frac{y \cdot y - x}{2 \cdot y} \]

2. Simplified0.0

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{x}{y} + y\right)} \]
   Proof

   [Start]18.0	\[ y - \frac{y \cdot y - x}{2 \cdot y} \]
   sub-neg [=>]18.0	\[ \color{blue}{y + \left(-\frac{y \cdot y - x}{2 \cdot y}\right)} \]
   +-commutative [=>]18.0	\[ \color{blue}{\left(-\frac{y \cdot y - x}{2 \cdot y}\right) + y} \]
   neg-sub0 [=>]18.0	\[ \color{blue}{\left(0 - \frac{y \cdot y - x}{2 \cdot y}\right)} + y \]
   associate-+l- [=>]18.0	\[ \color{blue}{0 - \left(\frac{y \cdot y - x}{2 \cdot y} - y\right)} \]
   sub0-neg [=>]18.0	\[ \color{blue}{-\left(\frac{y \cdot y - x}{2 \cdot y} - y\right)} \]
   div-sub [=>]18.0	\[ -\left(\color{blue}{\left(\frac{y \cdot y}{2 \cdot y} - \frac{x}{2 \cdot y}\right)} - y\right) \]
   sub-neg [=>]18.0	\[ -\left(\color{blue}{\left(\frac{y \cdot y}{2 \cdot y} + \left(-\frac{x}{2 \cdot y}\right)\right)} - y\right) \]
   +-commutative [=>]18.0	\[ -\left(\color{blue}{\left(\left(-\frac{x}{2 \cdot y}\right) + \frac{y \cdot y}{2 \cdot y}\right)} - y\right) \]
   times-frac [=>]0.0	\[ -\left(\left(\left(-\frac{x}{2 \cdot y}\right) + \color{blue}{\frac{y}{2} \cdot \frac{y}{y}}\right) - y\right) \]
   *-inverses [=>]0.0	\[ -\left(\left(\left(-\frac{x}{2 \cdot y}\right) + \frac{y}{2} \cdot \color{blue}{1}\right) - y\right) \]
   *-rgt-identity [=>]0.0	\[ -\left(\left(\left(-\frac{x}{2 \cdot y}\right) + \color{blue}{\frac{y}{2}}\right) - y\right) \]
   associate--l+ [=>]0.0	\[ -\color{blue}{\left(\left(-\frac{x}{2 \cdot y}\right) + \left(\frac{y}{2} - y\right)\right)} \]
   +-commutative [=>]0.0	\[ -\color{blue}{\left(\left(\frac{y}{2} - y\right) + \left(-\frac{x}{2 \cdot y}\right)\right)} \]

3. Final simplification0.0

   \[\leadsto 0.5 \cdot \left(y + \frac{x}{y}\right) \]

Alternatives

Error

Reproduce?

herbie shell --seed 1 
(FPCore (y x)
  :name "y - (y*y - x) / (2 y)"
  :precision binary64
  :pre (and (and (<= -1.79e+308 y) (<= y 1.79e+308)) (and (<= -1.79e+308 x) (<= x 1.79e+308)))
  (- y (/ (- (* y y) x) (* 2.0 y))))