Average Error: 0.5 → 0.4
Time: 9.8s
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 r303108 = x;
        double r303109 = 1.0;
        double r303110 = r303108 - r303109;
        double r303111 = sqrt(r303110);
        double r303112 = sqrt(r303108);
        double r303113 = r303111 * r303112;
        return r303113;
}
↓
double f(double x) {
        double r303114 = x;
        double r303115 = -0.5;
        double r303116 = -0.125;
        double r303117 = r303116 / r303114;
        double r303118 = r303115 + r303117;
        double r303119 = r303114 + r303118;
        return r303119;
}

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

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

  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)))