Average Error: 0.1 → 0.1
Time: 15.7s
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 r180180 = 1.0;
        double r180181 = x;
        double r180182 = r180180 - r180181;
        double r180183 = y;
        double r180184 = sqrt(r180181);
        double r180185 = r180183 * r180184;
        double r180186 = r180182 + r180185;
        return r180186;
}
↓
double f(double x, double y) {
        double r180187 = y;
        double r180188 = x;
        double r180189 = sqrt(r180188);
        double r180190 = 1.0;
        double r180191 = r180190 - r180188;
        double r180192 = fma(r180187, r180189, r180191);
        return r180192;
}

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 2019322 +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))))