Average Error: 0.1 → 0.1
Time: 4.1s
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 r84535 = 1.0;
        double r84536 = x;
        double r84537 = r84535 - r84536;
        double r84538 = y;
        double r84539 = sqrt(r84536);
        double r84540 = r84538 * r84539;
        double r84541 = r84537 + r84540;
        return r84541;
}
↓
double f(double x, double y) {
        double r84542 = x;
        double r84543 = sqrt(r84542);
        double r84544 = y;
        double r84545 = 1.0;
        double r84546 = r84545 - r84542;
        double r84547 = fma(r84543, r84544, r84546);
        return r84547;
}

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 2020060 +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))))