Average Error: 0.0 → 0.0
Time: 4.4s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[\begin{array}{l} t_0 := \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}\\ \mathsf{fma}\left(x - t_0, 1, t_0 - t_0\right) \end{array} \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (fma y (* x 0.5) 1.0)))) (fma (- x t_0) 1.0 (- t_0 t_0))))
double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}
↓
double code(double x, double y) {
	double t_0 = y / fma(y, (x * 0.5), 1.0);
	return fma((x - t_0), 1.0, (t_0 - t_0));
}

function code(x, y)
	return Float64(x - Float64(y / Float64(1.0 + Float64(Float64(x * y) / 2.0))))
end
↓
function code(x, y)
	t_0 = Float64(y / fma(y, Float64(x * 0.5), 1.0))
	return fma(Float64(x - t_0), 1.0, Float64(t_0 - t_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_] := Block[{t$95$0 = N[(y / N[(y * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(x - t$95$0), $MachinePrecision] * 1.0 + N[(t$95$0 - t$95$0), $MachinePrecision]), $MachinePrecision]]

x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
\begin{array}{l}
t_0 := \frac{y}{\mathsf{fma}\left(y, x \cdot 0.5, 1\right)}\\
\mathsf{fma}\left(x - t_0, 1, t_0 - t_0\right)
\end{array}

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. Taylor expanded in x around 0 0.0

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

  3. Simplified0.0

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

  4. Applied egg-rr0.1

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

  5. Applied egg-rr0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022153 
(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)))))