Average Error: 0.1 → 0.1
Time: 18.7s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x
double f(double x, double y) {
        double r74035 = 1.0;
        double r74036 = x;
        double r74037 = r74035 - r74036;
        double r74038 = y;
        double r74039 = sqrt(r74036);
        double r74040 = r74038 * r74039;
        double r74041 = r74037 + r74040;
        return r74041;
}
↓
double f(double x, double y) {
        double r74042 = y;
        double r74043 = x;
        double r74044 = sqrt(r74043);
        double r74045 = 1.0;
        double r74046 = fma(r74042, r74044, r74045);
        double r74047 = r74046 - r74043;
        return r74047;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))