Average Error: 0.0 → 0.0
Time: 2.7s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[\mathsf{fma}\left(-1, \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}, x\right) \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (fma -1.0 (/ y (fma y (* x 0.5) 1.0)) x))
double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}
↓
double code(double x, double y) {
	return fma(-1.0, (y / fma(y, (x * 0.5), 1.0)), x);
}

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end
↓
function code(x, y)
	return fma(-1.0, Float64(y / fma(y, Float64(x * 0.5), 1.0)), x)
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[(-1.0 * N[(y / N[(y * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
\mathsf{fma}\left(-1, \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}, x\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-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022166 
(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)))))