Average Error: 0.5 → 0.5
Time: 8.1s
Precision: 64
\[\sqrt{x - 1} \cdot \sqrt{x}\]
↓
\[x - \left(0.5 + \frac{0.125}{x}\right)\]
\sqrt{x - 1} \cdot \sqrt{x}
↓
x - \left(0.5 + \frac{0.125}{x}\right)
double f(double x) {
        double r10966 = x;
        double r10967 = 1.0;
        double r10968 = r10966 - r10967;
        double r10969 = sqrt(r10968);
        double r10970 = sqrt(r10966);
        double r10971 = r10969 * r10970;
        return r10971;
}
↓
double f(double x) {
        double r10972 = x;
        double r10973 = 0.5;
        double r10974 = 0.125;
        double r10975 = r10974 / r10972;
        double r10976 = r10973 + r10975;
        double r10977 = r10972 - r10976;
        return r10977;
}

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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))