Average Error: 0.1 → 0.1
Time: 4.8s
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 r74116 = 1.0;
        double r74117 = x;
        double r74118 = r74116 - r74117;
        double r74119 = y;
        double r74120 = sqrt(r74117);
        double r74121 = r74119 * r74120;
        double r74122 = r74118 + r74121;
        return r74122;
}
↓
double f(double x, double y) {
        double r74123 = x;
        double r74124 = sqrt(r74123);
        double r74125 = y;
        double r74126 = 1.0;
        double r74127 = r74126 - r74123;
        double r74128 = fma(r74124, r74125, r74127);
        return r74128;
}

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 2020036 +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))))