Average Error: 0.1 → 0.1
Time: 20.2s
Precision: 64
\[\left(1.0 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x\]
\left(1.0 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x
double f(double x, double y) {
        double r3477002 = 1.0;
        double r3477003 = x;
        double r3477004 = r3477002 - r3477003;
        double r3477005 = y;
        double r3477006 = sqrt(r3477003);
        double r3477007 = r3477005 * r3477006;
        double r3477008 = r3477004 + r3477007;
        return r3477008;
}
↓
double f(double x, double y) {
        double r3477009 = y;
        double r3477010 = x;
        double r3477011 = sqrt(r3477010);
        double r3477012 = 1.0;
        double r3477013 = fma(r3477009, r3477011, r3477012);
        double r3477014 = r3477013 - r3477010;
        return r3477014;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1.0 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))