Average Error: 0.5 → 0.4
Time: 3.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)
double f(double x) {
        double r15737 = x;
        double r15738 = 1.0;
        double r15739 = r15737 - r15738;
        double r15740 = sqrt(r15739);
        double r15741 = sqrt(r15737);
        double r15742 = r15740 * r15741;
        return r15742;
}
↓
double f(double x) {
        double r15743 = x;
        double r15744 = 0.125;
        double r15745 = 1.0;
        double r15746 = r15745 / r15743;
        double r15747 = 0.5;
        double r15748 = fma(r15744, r15746, r15747);
        double r15749 = r15743 - r15748;
        return r15749;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.5

    \[\sqrt{x - 1} \cdot \sqrt{x}\]

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)}\]

  4. Final simplification0.4

    \[\leadsto x - \mathsf{fma}\left(0.125, \frac{1}{x}, 0.5\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))