Average Error: 0.1 → 0.1
Time: 4.3s
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 r88421 = 1.0;
        double r88422 = x;
        double r88423 = r88421 - r88422;
        double r88424 = y;
        double r88425 = sqrt(r88422);
        double r88426 = r88424 * r88425;
        double r88427 = r88423 + r88426;
        return r88427;
}
↓
double f(double x, double y) {
        double r88428 = x;
        double r88429 = sqrt(r88428);
        double r88430 = y;
        double r88431 = 1.0;
        double r88432 = r88431 - r88428;
        double r88433 = fma(r88429, r88430, r88432);
        return r88433;
}

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 2020036 +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))))