Average Error: 0.0 → 0.0
Time: 2.4s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[x - \frac{-1}{\mathsf{fma}\left(x, -0.5, \frac{-1}{y}\right)} \]
(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (/ -1.0 (fma x -0.5 (/ -1.0 y)))))
double code(double x, double y) {
	return x - (y / (1.0 + ((x * y) / 2.0)));
}
↓
double code(double x, double y) {
	return x - (-1.0 / fma(x, -0.5, (-1.0 / y)));
}

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(-1.0 / fma(x, -0.5, Float64(-1.0 / y))))
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[(-1.0 / N[(x * -0.5 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022210 
(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)))))