Average Error: 0.5 → 0.5
Time: 8.0s
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 r332940 = x;
        double r332941 = 1.0;
        double r332942 = r332940 - r332941;
        double r332943 = sqrt(r332942);
        double r332944 = sqrt(r332940);
        double r332945 = r332943 * r332944;
        return r332945;
}
↓
double f(double x) {
        double r332946 = x;
        double r332947 = 0.125;
        double r332948 = r332947 / r332946;
        double r332949 = r332946 - r332948;
        double r332950 = 0.5;
        double r332951 = r332949 - r332950;
        return r332951;
}

Error

Bits error versus x

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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 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))