Average Error: 0.5 → 0.4
Time: 22.0s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)
double f(double x) {
        double r903041 = x;
        double r903042 = 1.0;
        double r903043 = r903041 - r903042;
        double r903044 = sqrt(r903043);
        double r903045 = sqrt(r903041);
        double r903046 = r903044 * r903045;
        return r903046;
}
↓
double f(double x) {
        double r903047 = x;
        double r903048 = 0.5;
        double r903049 = -0.125;
        double r903050 = r903049 / r903047;
        double r903051 = r903048 - r903050;
        double r903052 = r903047 - r903051;
        return r903052;
}

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

    \[\leadsto \color{blue}{x - \left(\frac{1}{8} \cdot \frac{1}{x} + \frac{1}{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)}\]

  4. Final simplification0.4

    \[\leadsto x - \left(\frac{1}{2} - \frac{\frac{-1}{8}}{x}\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1)) (sqrt x)))