Average Error: 0.0 → 0.1
Time: 3.0s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[x - {\left(\mathsf{fma}\left(0.5, x, \frac{1}{y}\right)\right)}^{-1} \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (pow (fma 0.5 x (/ 1.0 y)) -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 - pow(fma(0.5, x, (1.0 / y)), -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 - (fma(0.5, x, Float64(1.0 / y)) ^ -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[Power[N[(0.5 * x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

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

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. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}} \]

  3. Applied egg-rr0.1

    \[\leadsto x - \color{blue}{{\left(\frac{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)}{y}\right)}^{-1}} \]

  4. Taylor expanded in x around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022133 
(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)))))