Average Error: 0.5 → 0.4
Time: 11.1s
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 r273936 = x;
        double r273937 = 1.0;
        double r273938 = r273936 - r273937;
        double r273939 = sqrt(r273938);
        double r273940 = sqrt(r273936);
        double r273941 = r273939 * r273940;
        return r273941;
}
↓
double f(double x) {
        double r273942 = x;
        double r273943 = 0.125;
        double r273944 = r273943 / r273942;
        double r273945 = r273942 - r273944;
        double r273946 = 0.5;
        double r273947 = r273945 - r273946;
        return r273947;
}

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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))