Average Error: 0.5 → 0.4
Time: 2.8s
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 r8577 = x;
        double r8578 = 1.0;
        double r8579 = r8577 - r8578;
        double r8580 = sqrt(r8579);
        double r8581 = sqrt(r8577);
        double r8582 = r8580 * r8581;
        return r8582;
}
↓
double f(double x) {
        double r8583 = x;
        double r8584 = 0.125;
        double r8585 = 1.0;
        double r8586 = r8585 / r8583;
        double r8587 = 0.5;
        double r8588 = fma(r8584, r8586, r8587);
        double r8589 = r8583 - r8588;
        return r8589;
}

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 2020025 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))