Average Error: 0.1 → 0.1
Time: 3.3m
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x
double f(double x, double y) {
        double r12349896 = 1.0;
        double r12349897 = x;
        double r12349898 = r12349896 - r12349897;
        double r12349899 = y;
        double r12349900 = sqrt(r12349897);
        double r12349901 = r12349899 * r12349900;
        double r12349902 = r12349898 + r12349901;
        return r12349902;
}
↓
double f(double x, double y) {
        double r12349903 = y;
        double r12349904 = x;
        double r12349905 = sqrt(r12349904);
        double r12349906 = 1.0;
        double r12349907 = fma(r12349903, r12349905, r12349906);
        double r12349908 = r12349907 - r12349904;
        return r12349908;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1\right) - x\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))