Average Error: 0.0 → 0.1
Time: 1.5s
Precision: 64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)} \cdot y\]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)} \cdot y
double code(double x, double y) {
	return ((double) (x - ((double) (y / ((double) (1.0 + ((double) (((double) (x * y)) / 2.0))))))));
}
↓
double code(double x, double y) {
	return ((double) (x - ((double) (((double) (1.0 / ((double) fma(((double) (x / 2.0)), y, 1.0)))) * y))));
}

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 0.0

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

  3. Applied clear-num0.1

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

  5. Applied div-inv0.1

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

  6. Applied add-cube-cbrt0.1

    \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(1 + \frac{x \cdot y}{2}\right) \cdot \frac{1}{y}}\]

  7. Applied times-frac0.1

    \[\leadsto x - \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1 + \frac{x \cdot y}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}}\]

  8. Simplified0.1

    \[\leadsto x - \color{blue}{\frac{1}{\mathsf{fma}\left(\frac{x}{2}, y, 1\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}\]

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020123 +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)))))