Average Error: 0.1 → 0.1
Time: 9.9s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)
double f(double x, double y) {
        double r113271 = 1.0;
        double r113272 = x;
        double r113273 = r113271 - r113272;
        double r113274 = y;
        double r113275 = sqrt(r113272);
        double r113276 = r113274 * r113275;
        double r113277 = r113273 + r113276;
        return r113277;
}
↓
double f(double x, double y) {
        double r113278 = y;
        double r113279 = x;
        double r113280 = sqrt(r113279);
        double r113281 = 1.0;
        double r113282 = r113281 - r113279;
        double r113283 = fma(r113278, r113280, r113282);
        return r113283;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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))))