Average Error: 0.0 → 0.0
Time: 2.6s
Precision: binary64
Cost: 6976
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[x - \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)} \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (/ y (fma y (* x 0.5) 1.0))))
double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}
↓
double code(double x, double y) {
	return x - (y / fma(y, (x * 0.5), 1.0));
}

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end
↓
function code(x, y)
	return Float64(x - Float64(y / fma(y, Float64(x * 0.5), 1.0)))
end

code[x_, y_] := N[(x - N[(y / N[(1.0 + N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(x - N[(y / N[(y * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}

Error

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]

  2. Taylor expanded in x around 0 0.0

    \[\leadsto x - \frac{y}{\color{blue}{1 + 0.5 \cdot \left(y \cdot x\right)}} \]

  3. Simplified0.0

    \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}} \]

  4. Final simplification0.0

    \[\leadsto x - \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)} \]

Alternatives

Alternative 1
Error0.0
Cost704
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
Alternative 2
Error5.7
Cost584
\[\begin{array}{l} t_0 := x + \frac{-2}{x}\\ \mathbf{if}\;y \leq -9.451573621236853 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.553450586090769 \cdot 10^{+97}:\\ \;\;\;\;x - y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 3
Error15.7
Cost192
\[x - y \]

Error

Reproduce

herbie shell --seed 2022216 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))