Average Error: 0.1 → 0.1
Time: 3.7s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(\sqrt{x}, y, 1 - x\right)
double f(double x, double y) {
        double r74243 = 1.0;
        double r74244 = x;
        double r74245 = r74243 - r74244;
        double r74246 = y;
        double r74247 = sqrt(r74244);
        double r74248 = r74246 * r74247;
        double r74249 = r74245 + r74248;
        return r74249;
}
↓
double f(double x, double y) {
        double r74250 = x;
        double r74251 = sqrt(r74250);
        double r74252 = y;
        double r74253 = 1.0;
        double r74254 = r74253 - r74250;
        double r74255 = fma(r74251, r74252, r74254);
        return r74255;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1 x) (* y (sqrt x))))