Average Error: 0.5 → 0.3
Time: 10.9s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[\left(x - \frac{0.125}{x}\right) - 0.5\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
\left(x - \frac{0.125}{x}\right) - 0.5
double f(double x) {
        double r273899 = x;
        double r273900 = 1.0;
        double r273901 = r273899 - r273900;
        double r273902 = sqrt(r273901);
        double r273903 = sqrt(r273899);
        double r273904 = r273902 * r273903;
        return r273904;
}
↓
double f(double x) {
        double r273905 = x;
        double r273906 = 0.125;
        double r273907 = r273906 / r273905;
        double r273908 = r273905 - r273907;
        double r273909 = 0.5;
        double r273910 = r273908 - r273909;
        return r273910;
}

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 0.3

    \[\leadsto \color{blue}{x - \left(0.125 \cdot \frac{1}{x} + 0.5\right)}\]

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))