Average Error: 0.0 → 0.0
Time: 2.9s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[x - \frac{y}{\mathsf{fma}\left(x, y \cdot 0.5, 1\right)} \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (/ y (fma x (* y 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(x, (y * 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(x, Float64(y * 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[(x * N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr9.0

    \[\leadsto x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\mathsf{fma}\left(y \cdot x, 0.5, 1\right)}\right)\right)} \]

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022150 
(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)))))