Average Error: 0.1 → 0.1
Time: 16.6s
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 r93536 = 1.0;
        double r93537 = x;
        double r93538 = r93536 - r93537;
        double r93539 = y;
        double r93540 = sqrt(r93537);
        double r93541 = r93539 * r93540;
        double r93542 = r93538 + r93541;
        return r93542;
}
↓
double f(double x, double y) {
        double r93543 = y;
        double r93544 = x;
        double r93545 = sqrt(r93544);
        double r93546 = 1.0;
        double r93547 = r93546 - r93544;
        double r93548 = fma(r93543, r93545, r93547);
        return r93548;
}

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 2019303 +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))))