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 r104030 = 1.0;
        double r104031 = x;
        double r104032 = r104030 - r104031;
        double r104033 = y;
        double r104034 = sqrt(r104031);
        double r104035 = r104033 * r104034;
        double r104036 = r104032 + r104035;
        return r104036;
}
↓
double f(double x, double y) {
        double r104037 = x;
        double r104038 = sqrt(r104037);
        double r104039 = y;
        double r104040 = 1.0;
        double r104041 = r104040 - r104037;
        double r104042 = fma(r104038, r104039, r104041);
        return r104042;
}

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 2020062 +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))))