Average Error: 0.0 → 0.1
Time: 3.3s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{y}{1 + \frac{\frac{x}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \frac{y}{\sqrt{2}}}\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{1 + \frac{\frac{x}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \frac{y}{\sqrt{2}}}
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 + (((x / sqrt(sqrt(2.0))) / sqrt(sqrt(2.0))) * (y / sqrt(2.0))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.1

    \[\leadsto x - \frac{y}{1 + \frac{x \cdot y}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}\]

  4. Applied times-frac0.1

    \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{x}{\sqrt{2}} \cdot \frac{y}{\sqrt{2}}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.1

    \[\leadsto x - \frac{y}{1 + \frac{x}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \frac{y}{\sqrt{2}}}\]

  7. Applied sqrt-prod0.1

    \[\leadsto x - \frac{y}{1 + \frac{x}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \frac{y}{\sqrt{2}}}\]

  8. Applied associate-/r*0.1

    \[\leadsto x - \frac{y}{1 + \color{blue}{\frac{\frac{x}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}} \cdot \frac{y}{\sqrt{2}}}\]

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  :precision binary64
  (- x (/ y (+ 1 (/ (* x y) 2)))))