Average Error: 0.0 → 0.1
Time: 16.2s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1.0}{y}\right)}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1.0}{y}\right)}
double f(double x, double y) {
        double r12273369 = x;
        double r12273370 = y;
        double r12273371 = 1.0;
        double r12273372 = r12273369 * r12273370;
        double r12273373 = 2.0;
        double r12273374 = r12273372 / r12273373;
        double r12273375 = r12273371 + r12273374;
        double r12273376 = r12273370 / r12273375;
        double r12273377 = r12273369 - r12273376;
        return r12273377;
}
↓
double f(double x, double y) {
        double r12273378 = x;
        double r12273379 = 1.0;
        double r12273380 = 0.5;
        double r12273381 = 1.0;
        double r12273382 = y;
        double r12273383 = r12273381 / r12273382;
        double r12273384 = fma(r12273378, r12273380, r12273383);
        double r12273385 = r12273379 / r12273384;
        double r12273386 = r12273378 - r12273385;
        return r12273386;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}}\]
  3. Using strategy rm

  4. Applied clear-num0.1

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{x}{2.0}, y, 1.0\right)}{y}}}\]

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{0.5 \cdot x + 1.0 \cdot \frac{1}{y}}}\]

  6. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\mathsf{fma}\left(x, 0.5, \frac{1.0}{y}\right)}}\]

  7. Final simplification0.1

    \[\leadsto x - \frac{1}{\mathsf{fma}\left(x, 0.5, \frac{1.0}{y}\right)}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))