Average Error: 0.1 → 0.1
Time: 4.5s
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 r78204 = 1.0;
        double r78205 = x;
        double r78206 = r78204 - r78205;
        double r78207 = y;
        double r78208 = sqrt(r78205);
        double r78209 = r78207 * r78208;
        double r78210 = r78206 + r78209;
        return r78210;
}
↓
double f(double x, double y) {
        double r78211 = x;
        double r78212 = sqrt(r78211);
        double r78213 = y;
        double r78214 = 1.0;
        double r78215 = r78214 - r78211;
        double r78216 = fma(r78212, r78213, r78215);
        return r78216;
}

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))))