Average Error: 0.1 → 0.1
Time: 4.7s
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 r91836 = 1.0;
        double r91837 = x;
        double r91838 = r91836 - r91837;
        double r91839 = y;
        double r91840 = sqrt(r91837);
        double r91841 = r91839 * r91840;
        double r91842 = r91838 + r91841;
        return r91842;
}
↓
double f(double x, double y) {
        double r91843 = x;
        double r91844 = sqrt(r91843);
        double r91845 = y;
        double r91846 = 1.0;
        double r91847 = r91846 - r91843;
        double r91848 = fma(r91844, r91845, r91847);
        return r91848;
}

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 2020001 +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))))