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 r100033 = 1.0;
        double r100034 = x;
        double r100035 = r100033 - r100034;
        double r100036 = y;
        double r100037 = sqrt(r100034);
        double r100038 = r100036 * r100037;
        double r100039 = r100035 + r100038;
        return r100039;
}
↓
double f(double x, double y) {
        double r100040 = x;
        double r100041 = sqrt(r100040);
        double r100042 = y;
        double r100043 = 1.0;
        double r100044 = r100043 - r100040;
        double r100045 = fma(r100041, r100042, r100044);
        return r100045;
}

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))))