Average Error: 0.1 → 0.1
Time: 18.0s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)
double f(double x, double y) {
        double r91813 = 1.0;
        double r91814 = x;
        double r91815 = r91813 - r91814;
        double r91816 = y;
        double r91817 = sqrt(r91814);
        double r91818 = r91816 * r91817;
        double r91819 = r91815 + r91818;
        return r91819;
}
↓
double f(double x, double y) {
        double r91820 = y;
        double r91821 = x;
        double r91822 = sqrt(r91821);
        double r91823 = 1.0;
        double r91824 = r91823 - r91821;
        double r91825 = fma(r91820, r91822, r91824);
        return r91825;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))