Average Error: 0.5 → 0.4
Time: 12.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - 0.5\right) - \frac{0.125}{x}\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - 0.5\right) - \frac{0.125}{x}
double f(double x) {
        double r10901 = x;
        double r10902 = 1.0;
        double r10903 = r10901 - r10902;
        double r10904 = sqrt(r10903);
        double r10905 = sqrt(r10901);
        double r10906 = r10904 * r10905;
        return r10906;
}
↓
double f(double x) {
        double r10907 = x;
        double r10908 = 0.5;
        double r10909 = r10907 - r10908;
        double r10910 = 0.125;
        double r10911 = r10910 / r10907;
        double r10912 = r10909 - r10911;
        return r10912;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{\left(x - 0.5\right) - \frac{0.125}{x}}\]

  4. Final simplification0.4

    \[\leadsto \left(x - 0.5\right) - \frac{0.125}{x}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))