Average Error: 0.1 → 0.1
Time: 6.8s
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 r435 = 1.0;
        double r436 = x;
        double r437 = r435 - r436;
        double r438 = y;
        double r439 = sqrt(r436);
        double r440 = r438 * r439;
        double r441 = r437 + r440;
        return r441;
}
↓
double f(double x, double y) {
        double r442 = x;
        double r443 = sqrt(r442);
        double r444 = y;
        double r445 = 1.0;
        double r446 = r445 - r442;
        double r447 = fma(r443, r444, r446);
        return r447;
}

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 2020025 +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))))