Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[x - y \cdot \frac{1}{\mathsf{fma}\left(x, \frac{y}{2}, 1\right)} \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (* y (/ 1.0 (fma x (/ y 2.0) 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 * (1.0 / fma(x, (y / 2.0), 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 * Float64(1.0 / fma(x, Float64(y / 2.0), 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[(1.0 / N[(x * N[(y / 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - y \cdot \frac{1}{\mathsf{fma}\left(x, \frac{y}{2}, 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. Simplified0.0

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

  3. Applied div-inv_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022131 
(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)))))