Average Error: 0.1 → 0.1
Time: 24.5s
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 r82423 = 1.0;
        double r82424 = x;
        double r82425 = r82423 - r82424;
        double r82426 = y;
        double r82427 = sqrt(r82424);
        double r82428 = r82426 * r82427;
        double r82429 = r82425 + r82428;
        return r82429;
}
↓
double f(double x, double y) {
        double r82430 = y;
        double r82431 = x;
        double r82432 = sqrt(r82431);
        double r82433 = 1.0;
        double r82434 = r82433 - r82431;
        double r82435 = fma(r82430, r82432, r82434);
        return r82435;
}

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 2019199 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))