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 r15311 = x;
        double r15312 = 1.0;
        double r15313 = r15311 - r15312;
        double r15314 = sqrt(r15313);
        double r15315 = sqrt(r15311);
        double r15316 = r15314 * r15315;
        return r15316;
}
↓
double f(double x) {
        double r15317 = x;
        double r15318 = 0.125;
        double r15319 = 1.0;
        double r15320 = r15319 / r15317;
        double r15321 = 0.5;
        double r15322 = fma(r15318, r15320, r15321);
        double r15323 = r15317 - r15322;
        return r15323;
}

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)))