Average Error: 0.5 → 0.4
Time: 2.4s
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 r1569 = x;
        double r1570 = 1.0;
        double r1571 = r1569 - r1570;
        double r1572 = sqrt(r1571);
        double r1573 = sqrt(r1569);
        double r1574 = r1572 * r1573;
        return r1574;
}
↓
double f(double x) {
        double r1575 = x;
        double r1576 = 0.125;
        double r1577 = 1.0;
        double r1578 = r1577 / r1575;
        double r1579 = 0.5;
        double r1580 = fma(r1576, r1578, r1579);
        double r1581 = r1575 - r1580;
        return r1581;
}

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 2020034 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))