Average Error: 0.1 → 0.1
Time: 4.3s
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 r95273 = 1.0;
        double r95274 = x;
        double r95275 = r95273 - r95274;
        double r95276 = y;
        double r95277 = sqrt(r95274);
        double r95278 = r95276 * r95277;
        double r95279 = r95275 + r95278;
        return r95279;
}
↓
double f(double x, double y) {
        double r95280 = x;
        double r95281 = sqrt(r95280);
        double r95282 = y;
        double r95283 = 1.0;
        double r95284 = r95283 - r95280;
        double r95285 = fma(r95281, r95282, r95284);
        return r95285;
}

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 2020039 +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))))