Average Error: 0.1 → 0.1
Time: 11.0s
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 r84565 = 1.0;
        double r84566 = x;
        double r84567 = r84565 - r84566;
        double r84568 = y;
        double r84569 = sqrt(r84566);
        double r84570 = r84568 * r84569;
        double r84571 = r84567 + r84570;
        return r84571;
}
↓
double f(double x, double y) {
        double r84572 = y;
        double r84573 = x;
        double r84574 = sqrt(r84573);
        double r84575 = 1.0;
        double r84576 = r84575 - r84573;
        double r84577 = fma(r84572, r84574, r84576);
        return r84577;
}

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 2020043 +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))))