Average Error: 0.5 → 0.5
Time: 8.3s
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 r12073 = x;
        double r12074 = 1.0;
        double r12075 = r12073 - r12074;
        double r12076 = sqrt(r12075);
        double r12077 = sqrt(r12073);
        double r12078 = r12076 * r12077;
        return r12078;
}
↓
double f(double x) {
        double r12079 = x;
        double r12080 = 0.125;
        double r12081 = r12080 / r12079;
        double r12082 = r12079 - r12081;
        double r12083 = 0.5;
        double r12084 = r12082 - r12083;
        return r12084;
}

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 2019195 +o rules:numerics
(FPCore (x)
  :name "sqrt times"
  (* (sqrt (- x 1.0)) (sqrt x)))