Average Error: 0.1 → 0.1
Time: 4.4s
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 r100029 = 1.0;
        double r100030 = x;
        double r100031 = r100029 - r100030;
        double r100032 = y;
        double r100033 = sqrt(r100030);
        double r100034 = r100032 * r100033;
        double r100035 = r100031 + r100034;
        return r100035;
}
↓
double f(double x, double y) {
        double r100036 = x;
        double r100037 = sqrt(r100036);
        double r100038 = y;
        double r100039 = 1.0;
        double r100040 = r100039 - r100036;
        double r100041 = fma(r100037, r100038, r100040);
        return r100041;
}

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 2019353 +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))))