Average Error: 0.0 → 0.0
Time: 3.9s
Precision: binary64
\[x - \frac{y}{1 + \frac{x \cdot y}{2}} \]
↓
\[x - \frac{y}{\mathsf{fma}\left(x, y \cdot 0.5, 1\right)} \]
x - \frac{y}{1 + \frac{x \cdot y}{2}}
↓
x - \frac{y}{\mathsf{fma}\left(x, y \cdot 0.5, 1\right)}

(FPCore (x y) :precision binary64 (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))
↓
(FPCore (x y) :precision binary64 (- x (/ y (fma x (* y 0.5) 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 / fma(x, (y * 0.5), 1.0));
}

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 clear-num_binary640.1

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

  4. Applied *-un-lft-identity_binary640.1

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

  5. Applied *-un-lft-identity_binary640.1

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

  6. Applied times-frac_binary640.1

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

  7. Applied add-cube-cbrt_binary640.1

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

  8. Applied times-frac_binary640.1

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

  9. Simplified0.1

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2022077 
(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)))))