Average Error: 0.5 → 0.4
Time: 1.8s
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 r1193 = x;
        double r1194 = 1.0;
        double r1195 = r1193 - r1194;
        double r1196 = sqrt(r1195);
        double r1197 = sqrt(r1193);
        double r1198 = r1196 * r1197;
        return r1198;
}
↓
double f(double x) {
        double r1199 = x;
        double r1200 = 0.125;
        double r1201 = 1.0;
        double r1202 = r1201 / r1199;
        double r1203 = 0.5;
        double r1204 = fma(r1200, r1202, r1203);
        double r1205 = r1199 - r1204;
        return r1205;
}

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 2020024 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))