Average Error: 0.1 → 0.1
Time: 4.7s
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 r99154 = 1.0;
        double r99155 = x;
        double r99156 = r99154 - r99155;
        double r99157 = y;
        double r99158 = sqrt(r99155);
        double r99159 = r99157 * r99158;
        double r99160 = r99156 + r99159;
        return r99160;
}
↓
double f(double x, double y) {
        double r99161 = x;
        double r99162 = sqrt(r99161);
        double r99163 = y;
        double r99164 = 1.0;
        double r99165 = r99164 - r99161;
        double r99166 = fma(r99162, r99163, r99165);
        return r99166;
}

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 2020083 +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))))