Average Error: 0.5 → 0.3
Time: 8.3s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(\frac{0.125}{x} + 0.5\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(\frac{0.125}{x} + 0.5\right)
double f(double x) {
        double r9863 = x;
        double r9864 = 1.0;
        double r9865 = r9863 - r9864;
        double r9866 = sqrt(r9865);
        double r9867 = sqrt(r9863);
        double r9868 = r9866 * r9867;
        return r9868;
}
↓
double f(double x) {
        double r9869 = x;
        double r9870 = 0.125;
        double r9871 = r9870 / r9869;
        double r9872 = 0.5;
        double r9873 = r9871 + r9872;
        double r9874 = r9869 - r9873;
        return r9874;
}

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

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))