Average Error: 0.5 → 0.4
Time: 2.1s
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 r3138 = x;
        double r3139 = 1.0;
        double r3140 = r3138 - r3139;
        double r3141 = sqrt(r3140);
        double r3142 = sqrt(r3138);
        double r3143 = r3141 * r3142;
        return r3143;
}
↓
double f(double x) {
        double r3144 = x;
        double r3145 = 0.125;
        double r3146 = 1.0;
        double r3147 = r3146 / r3144;
        double r3148 = 0.5;
        double r3149 = fma(r3145, r3147, r3148);
        double r3150 = r3144 - r3149;
        return r3150;
}

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 2020064 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))