Average Error: 0.1 → 0.1
Time: 17.4s
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 r72089 = 1.0;
        double r72090 = x;
        double r72091 = r72089 - r72090;
        double r72092 = y;
        double r72093 = sqrt(r72090);
        double r72094 = r72092 * r72093;
        double r72095 = r72091 + r72094;
        return r72095;
}
↓
double f(double x, double y) {
        double r72096 = y;
        double r72097 = x;
        double r72098 = sqrt(r72097);
        double r72099 = 1.0;
        double r72100 = r72099 - r72097;
        double r72101 = fma(r72096, r72098, r72100);
        return r72101;
}

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 2019305 +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))))